What is the value of ln(tanh(3-4i)) ?

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ln(tanh(3-4i))

Solution

ln (tanh (3 - 4 i))

Solution

ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

Solution steps

ln (tanh (3 - 4 i))

Rewrite using trig identities:

tanh (3 - 4 i) = \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

tanh (3 - 4 i)

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

= \frac{e ^{3 - 4 i} - e ^{- (3 - 4 i)}}{e ^{3 - 4 i} + e ^{- (3 - 4 i)}}

Simplify

\frac{e ^{3 - 4 i} - e ^{- (3 - 4 i)}}{e ^{3 - 4 i} + e ^{- (3 - 4 i)}} : \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )}

\frac{e ^{3 - 4 i} - e ^{- (3 - 4 i)}}{e ^{3 - 4 i} + e ^{- (3 - 4 i)}}

e^{3 - 4 i} - e^{- (3 - 4 i)} = e^{3} (cos (- 4) + i sin (- 4)) - e^{- 3} (cos (4) + i sin (4))

e^{3 - 4 i} - e^{- (3 - 4 i)}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{3} (cos (- 4) + i sin (- 4)) - e^{- (3 - 4 i)}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{3} (cos (- 4) + i sin (- 4)) - e^{- 3} (cos (4) + i sin (4))

= \frac{e ^{3} ( cos ( - 4 ) + i sin ( - 4 ) ) - e ^{- 3} ( cos ( 4 ) + i sin ( 4 ) )}{e ^{3 - 4 i} + e ^{- (3 - 4 i)}}

e^{3 - 4 i} + e^{- (3 - 4 i)} = e^{3} (cos (- 4) + i sin (- 4)) + e^{- 3} (cos (4) + i sin (4))

e^{3 - 4 i} + e^{- (3 - 4 i)}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{3} (cos (- 4) + i sin (- 4)) + e^{- (3 - 4 i)}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{3} (cos (- 4) + i sin (- 4)) + e^{- 3} (cos (4) + i sin (4))

= \frac{e ^{3} ( cos ( - 4 ) + i sin ( - 4 ) ) - e ^{- 3} ( cos ( 4 ) + i sin ( 4 ) )}{e ^{3} ( cos ( - 4 ) + i sin ( - 4 ) ) + e ^{- 3} ( cos ( 4 ) + i sin ( 4 ) )}

Apply complex arithmetic rule:

\frac{a + bi}{c + d i} = \frac{( c - d i ) ( a + bi )}{( c - d i ) ( c + d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}

a = \frac{e ^{6} cos ( - 4 ) - cos ( 4 )}{e ^{3}}, b = \frac{e ^{6} sin ( - 4 ) - sin ( 4 )}{e ^{3}}, c = \frac{e ^{6} cos ( - 4 ) + cos ( 4 )}{e ^{3}}, d = \frac{e ^{6} sin ( - 4 ) + sin ( 4 )}{e ^{3}}

= \frac{( \frac{e ^{6} c o s ( - 4 ) - c o s ( 4 )}{e ^{3}} \cdot \frac{e ^{6} c o s ( - 4 ) + c o s ( 4 )}{e ^{3}} + \frac{e ^{6} s i n ( - 4 ) - s i n ( 4 )}{e ^{3}} \cdot \frac{e ^{6} s i n ( - 4 ) + s i n ( 4 )}{e ^{3}} ) + ( \frac{e ^{6} s i n ( - 4 ) - s i n ( 4 )}{e ^{3}} \cdot \frac{e ^{6} c o s ( - 4 ) + c o s ( 4 )}{e ^{3}} - \frac{e ^{6} c o s ( - 4 ) - c o s ( 4 )}{e ^{3}} \cdot \frac{e ^{6} s i n ( - 4 ) + s i n ( 4 )}{e ^{3}} ) i}{( \frac{e ^{6} c o s ( - 4 ) + c o s ( 4 )}{e ^{3}} ) ^{2} + ( \frac{e ^{6} s i n ( - 4 ) + s i n ( 4 )}{e ^{3}} ) ^{2}}

Refine

= \frac{\frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} + \frac{2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 )}{e ^{6}} i}{\frac{( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) ^{2} + ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) ) ^{2}}{e ^{6}}}

Simplify

\frac{\frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} + \frac{2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 )}{e ^{6}} i}{\frac{( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) ^{2} + ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) ) ^{2}}{e ^{6}}} : \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

\frac{\frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} + \frac{2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 )}{e ^{6}} i}{\frac{( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) ^{2} + ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) ) ^{2}}{e ^{6}}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{( \frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} + \frac{2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 )}{e ^{6}} i ) e ^{6}}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Multiply

\frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{6}} i : \frac{i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{e ^{6}}

\frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{6}} i

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) ) i}{e ^{6}}

= \frac{e ^{6} ( \frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} + \frac{i ( 2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 ) )}{e ^{6}} )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Combine the fractions

\frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) )}{e ^{6}} + \frac{i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{e ^{6}} : \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{e ^{6}}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{e ^{6}}

= \frac{e ^{6} ( \frac{i ( 2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 ) ) + ( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) )}{e ^{6}} )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Remove parentheses:

(a) = a

= \frac{\frac{( e ^{6} c o s ( - 4 ) - c o s ( 4 ) ) ( e ^{6} c o s ( - 4 ) + c o s ( 4 ) ) + ( e ^{6} s i n ( - 4 ) - s i n ( 4 ) ) ( e ^{6} s i n ( - 4 ) + s i n ( 4 ) ) + ( 2 e ^{6} c o s ( 4 ) s i n ( - 4 ) - 2 e ^{6} s i n ( 4 ) c o s ( - 4 ) ) i}{e ^{6}} e ^{6}}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Multiply

\frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) ) i}{e ^{6}} e^{6} : (e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4)) + (e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) + i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4))

\frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) ) i}{e ^{6}} e^{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( ( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) ) i ) e ^{6}}{e ^{6}}

Cancel the common factor:

e^{6}

= (e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4)) + (e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) + (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4)) i

= \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

= \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Rewrite

\frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

in standard complex form:

\frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} i

\frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{( e ^{6} cos ( - 4 ) + cos ( 4 ) ) ^{2} + ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) ^{2}}

Expand

(e^{6} cos (- 4) + cos (4))^{2} + (e^{6} sin (- 4) + sin (4))^{2} : e^{12} cos^{2} (- 4) + 2 e^{6} cos (4) cos (- 4) + cos^{2} (4) + e^{12} sin^{2} (- 4) + 2 e^{6} sin (4) sin (- 4) + sin^{2} (4)

(e^{6} cos (- 4) + cos (4))^{2} + (e^{6} sin (- 4) + sin (4))^{2}

(e^{6} cos (- 4) + cos (4))^{2} : e^{12} cos^{2} (- 4) + 2 e^{6} cos (4) cos (- 4) + cos^{2} (4)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = e^{6} cos (- 4), b = cos (4)

= (e^{6} cos (- 4))^{2} + 2 e^{6} cos (- 4) cos (4) + cos^{2} (4)

(e^{6} cos (- 4))^{2} = e^{12} cos^{2} (- 4)

(e^{6} cos (- 4))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= cos^{2} (- 4) (e^{6})^{2}

(e^{6})^{2} : e^{12}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= e^{12}

= e^{12} cos^{2} (- 4)

= e^{12} cos^{2} (- 4) + 2 e^{6} cos (4) cos (- 4) + cos^{2} (4)

= e^{12} cos^{2} (- 4) + 2 e^{6} cos (4) cos (- 4) + cos^{2} (4) + (e^{6} sin (- 4) + sin (4))^{2}

(e^{6} sin (- 4) + sin (4))^{2} : e^{12} sin^{2} (- 4) + 2 e^{6} sin (4) sin (- 4) + sin^{2} (4)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = e^{6} sin (- 4), b = sin (4)

= (e^{6} sin (- 4))^{2} + 2 e^{6} sin (- 4) sin (4) + sin^{2} (4)

(e^{6} sin (- 4))^{2} = e^{12} sin^{2} (- 4)

(e^{6} sin (- 4))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= sin^{2} (- 4) (e^{6})^{2}

(e^{6})^{2} : e^{12}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= e^{12}

= e^{12} sin^{2} (- 4)

= e^{12} sin^{2} (- 4) + 2 e^{6} sin (4) sin (- 4) + sin^{2} (4)

= e^{12} cos^{2} (- 4) + 2 e^{6} cos (4) cos (- 4) + cos^{2} (4) + e^{12} sin^{2} (- 4) + 2 e^{6} sin (4) sin (- 4) + sin^{2} (4)

= \frac{( e ^{6} cos ( - 4 ) - cos ( 4 ) ) ( e ^{6} cos ( - 4 ) + cos ( 4 ) ) + ( e ^{6} sin ( - 4 ) - sin ( 4 ) ) ( e ^{6} sin ( - 4 ) + sin ( 4 ) ) + i ( 2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 ) )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )}

Expand

(e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4)) + (e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) + i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4)) : e^{12} cos^{2} (- 4) - cos^{2} (4) + e^{12} sin^{2} (- 4) - sin^{2} (4) + 2 e^{6} i cos (4) sin (- 4) - 2 e^{6} i sin (4) cos (- 4)

(e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4)) + (e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) + i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4))

Expand

(e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4)) : e^{12} cos^{2} (- 4) - cos^{2} (4)

(e^{6} cos (- 4) - cos (4)) (e^{6} cos (- 4) + cos (4))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = e^{6} cos (- 4), b = cos (4)

= (e^{6} cos (- 4))^{2} - cos^{2} (4)

(e^{6} cos (- 4))^{2} = e^{12} cos^{2} (- 4)

(e^{6} cos (- 4))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= cos^{2} (- 4) (e^{6})^{2}

(e^{6})^{2} : e^{12}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= e^{12}

= e^{12} cos^{2} (- 4)

= e^{12} cos^{2} (- 4) - cos^{2} (4)

= e^{12} cos^{2} (- 4) - cos^{2} (4) + (e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) + i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4))

Expand

(e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4)) : e^{12} sin^{2} (- 4) - sin^{2} (4)

(e^{6} sin (- 4) - sin (4)) (e^{6} sin (- 4) + sin (4))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = e^{6} sin (- 4), b = sin (4)

= (e^{6} sin (- 4))^{2} - sin^{2} (4)

(e^{6} sin (- 4))^{2} = e^{12} sin^{2} (- 4)

(e^{6} sin (- 4))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= sin^{2} (- 4) (e^{6})^{2}

(e^{6})^{2} : e^{12}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= e^{12}

= e^{12} sin^{2} (- 4)

= e^{12} sin^{2} (- 4) - sin^{2} (4)

= e^{12} cos^{2} (- 4) - cos^{2} (4) + e^{12} sin^{2} (- 4) - sin^{2} (4) + i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4))

Expand

i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4)) : 2 e^{6} i cos (4) sin (- 4) - 2 e^{6} i sin (4) cos (- 4)

i (2 e^{6} cos (4) sin (- 4) - 2 e^{6} sin (4) cos (- 4))

Apply the distributive law:

a (b - c) = ab - a c

a = i, b = 2 e^{6} cos (4) sin (- 4), c = 2 e^{6} sin (4) cos (- 4)

= i 2 e^{6} cos (4) sin (- 4) - i 2 e^{6} sin (4) cos (- 4)

= 2 e^{6} i cos (4) sin (- 4) - 2 e^{6} i sin (4) cos (- 4)

= e^{12} cos^{2} (- 4) - cos^{2} (4) + e^{12} sin^{2} (- 4) - sin^{2} (4) + 2 e^{6} i cos (4) sin (- 4) - 2 e^{6} i sin (4) cos (- 4)

= \frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 ) + 2 e ^{6} i cos ( 4 ) sin ( - 4 ) - 2 e ^{6} i sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 ) + 2 e ^{6} i cos ( 4 ) sin ( - 4 ) - 2 e ^{6} i sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} = \frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} + \frac{2 e ^{6} i cos ( 4 ) sin ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )} - \frac{2 e ^{6} i sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + sin ^{2} ( 4 )}

= \frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{2 e ^{6} i cos ( 4 ) sin ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{2 e ^{6} i sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

Group the real part and the imaginary part of the complex number

= (\frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}) + (\frac{2 e ^{6} cos ( 4 ) sin ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}) i

\frac{2 e ^{6} cos ( 4 ) sin ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} = \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

\frac{2 e ^{6} cos ( 4 ) sin ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

= (\frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}) + \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} i

\frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} = \frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

\frac{e ^{12} cos ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{cos ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{e ^{12} sin ^{2} ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )}

= \frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} i

= \frac{e ^{12} cos ^{2} ( - 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( - 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} + \frac{2 e ^{6} cos ( 4 ) sin ( - 4 ) - 2 e ^{6} sin ( 4 ) cos ( - 4 )}{e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} i

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) sin ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( - 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) sin ( - 4 )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( - 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 ) + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} sin ^{2} ( - 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

sin (- x) = - sin (x)

sin (- 4) = - sin (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( - 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2}}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Use the following property:

cos (- x) = cos (x)

cos (- 4) = cos (4)

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2}}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )} + i \frac{- 2 e ^{6} sin ( 4 ) cos ( 4 ) + 2 e ^{6} cos ( 4 ) ( - sin ( 4 ) )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} ( - sin ( 4 ) ) ^{2} + 2 e ^{6} cos ( 4 ) cos ( 4 ) + 2 e ^{6} sin ( 4 ) ( - sin ( 4 ) )}

Simplify

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

= ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

Simplify

ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}) : ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

Multiply

i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} : \frac{4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

i \frac{4 e ^{6} cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 e ^{6} cos ( 4 ) sin ( 4 ) i}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

= ln (\frac{e ^{12} cos ^{2} ( 4 ) - cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - sin ^{2} ( 4 )}{e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )} - \frac{4 e ^{6} i cos ( 4 ) sin ( 4 )}{e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 ) + cos ^{2} ( 4 ) + sin ^{2} ( 4 )})

Join

\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - \frac{4 e ^{6} cos ( 4 ) sin ( 4 ) i}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} : \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )} - \frac{4 e ^{6} cos ( 4 ) sin ( 4 ) i}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} cos ( 4 ) sin ( 4 ) i}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )}

= ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} cos ( 4 ) sin ( 4 ) i}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

= ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

= ln (\frac{- cos ^{2} ( 4 ) - sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) - 4 e ^{6} i cos ( 4 ) sin ( 4 )}{cos ^{2} ( 4 ) + sin ^{2} ( 4 ) + e ^{12} cos ^{2} ( 4 ) + e ^{12} sin ^{2} ( 4 ) + 2 e ^{6} cos ^{2} ( 4 ) - 2 e ^{6} sin ^{2} ( 4 )})

Popular Examples

arccos(1.66)

arccos (1.66)

arctan((sqrt(2))/(-1))

arctan (\frac{2}{- 1})

2-sin(pi/2)

2 - sin (\frac{π}{2})

arctan(20/5)

arctan (\frac{20}{5})

tan(-12/5)

tan (- \frac{12}{5})

Frequently Asked Questions (FAQ)

What is the value of ln(tanh(3-4i)) ?
The value of ln(tanh(3-4i)) is