What is the value of (-csch^2(2))/(coth(2)) ?

The value of (-csch^2(2))/(coth(2)) is -(4e^4)/((e^4-1)(e^4+1))

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(-csch^2(2))/(coth(2))

Solution

\frac{- csch ^{2} ( 2 )}{coth ( 2 )}

Solution

- \frac{4 e ^{4}}{( e ^{4} - 1 ) ( e ^{4} + 1 )}

Decimal Notation

- 0.07328 \dots

Solution steps

\frac{- csch ^{2} ( 2 )}{coth ( 2 )}

Rewrite using trig identities:

csch (2) = \frac{2 e ^{2}}{e ^{4} - 1}

csch (2)

Use the Hyperbolic identity:

csch (x) = \frac{2}{e ^{x} - e ^{- x}}

= \frac{2}{e ^{2} - e ^{- 2}}

\frac{2}{e ^{2} - e ^{- 2}} = \frac{2 e ^{2}}{e ^{4} - 1}

\frac{2}{e ^{2} - e ^{- 2}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{2}{e ^{2} - \frac{1}{e ^{2}}}

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

e^{2} - \frac{1}{e ^{2}}

Convert element to fraction:

e^{2} = \frac{e ^{2} e ^{2}}{e ^{2}}

= \frac{e ^{2} e ^{2}}{e ^{2}} - \frac{1}{e ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2} e ^{2} - 1}{e ^{2}}

e^{2} e^{2} - 1 = e^{4} - 1

e^{2} e^{2} - 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

= \frac{e ^{4} - 1}{e ^{2}}

= \frac{2}{\frac{e ^{4} - 1}{e ^{2}}}

Apply the fraction rule:

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

= \frac{2 e ^{2}}{e ^{4} - 1}

= \frac{2 e ^{2}}{e ^{4} - 1}

Rewrite using trig identities:

coth (2) = \frac{e ^{4} + 1}{e ^{4} - 1}

coth (2)

Use the Hyperbolic identity:

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

= \frac{e ^{2} + e ^{- 2}}{e ^{2} - e ^{- 2}}

\frac{e ^{2} + e ^{- 2}}{e ^{2} - e ^{- 2}} = \frac{e ^{4} + 1}{e ^{4} - 1}

\frac{e ^{2} + e ^{- 2}}{e ^{2} - e ^{- 2}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

e^{- 2} = \frac{1}{e ^{2}}

= \frac{e ^{2} + \frac{1}{e ^{2}}}{e ^{2} - \frac{1}{e ^{2}}}

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

e^{2} - \frac{1}{e ^{2}}

Convert element to fraction:

e^{2} = \frac{e ^{2} e ^{2}}{e ^{2}}

= \frac{e ^{2} e ^{2}}{e ^{2}} - \frac{1}{e ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2} e ^{2} - 1}{e ^{2}}

e^{2} e^{2} - 1 = e^{4} - 1

e^{2} e^{2} - 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

= \frac{e ^{4} - 1}{e ^{2}}

= \frac{e ^{2} + \frac{1}{e ^{2}}}{\frac{e ^{4} - 1}{e ^{2}}}

Join

e^{2} + \frac{1}{e ^{2}} : \frac{e ^{4} + 1}{e ^{2}}

e^{2} + \frac{1}{e ^{2}}

Convert element to fraction:

e^{2} = \frac{e ^{2} e ^{2}}{e ^{2}}

= \frac{e ^{2} e ^{2}}{e ^{2}} + \frac{1}{e ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2} e ^{2} + 1}{e ^{2}}

e^{2} e^{2} + 1 = e^{4} + 1

e^{2} e^{2} + 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} + 1

= \frac{e ^{4} + 1}{e ^{2}}

= \frac{\frac{e ^{4} + 1}{e ^{2}}}{\frac{e ^{4} - 1}{e ^{2}}}

Divide fractions:

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

= \frac{( e ^{4} + 1 ) e ^{2}}{e ^{2} ( e ^{4} - 1 )}

Cancel the common factor:

e^{2}

= \frac{e ^{4} + 1}{e ^{4} - 1}

= \frac{e ^{4} + 1}{e ^{4} - 1}

= \frac{- ( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2}}{\frac{e ^{4} + 1}{e ^{4} - 1}}

Simplify

\frac{- ( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2}}{\frac{e ^{4} + 1}{e ^{4} - 1}} : - \frac{4 e ^{4}}{( e ^{4} - 1 ) ( e ^{4} + 1 )}

\frac{- ( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2}}{\frac{e ^{4} + 1}{e ^{4} - 1}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2}}{\frac{e ^{4} + 1}{e ^{4} - 1}}

Apply the fraction rule:

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

\frac{( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2}}{\frac{e ^{4} + 1}{e ^{4} - 1}} = \frac{( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2} ( e ^{4} - 1 )}{e ^{4} + 1}

= - \frac{( \frac{2 e ^{2}}{e ^{4} - 1} ) ^{2} ( e ^{4} - 1 )}{e ^{4} + 1}

(\frac{2 e ^{2}}{e ^{4} - 1})^{2} = \frac{2 ^{2} e ^{4}}{( e ^{4} - 1 ) ^{2}}

(\frac{2 e ^{2}}{e ^{4} - 1})^{2}

Apply exponent rule:

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

= \frac{( 2 e ^{2} ) ^{2}}{( e ^{4} - 1 ) ^{2}}

Apply exponent rule:

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

(2 e^{2})^{2} = 2^{2} (e^{2})^{2}

= \frac{2 ^{2} ( e ^{2} ) ^{2}}{( e ^{4} - 1 ) ^{2}}

(e^{2})^{2} : e^{4}

Apply exponent rule:

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

= e^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= e^{4}

= \frac{2 ^{2} e ^{4}}{( e ^{4} - 1 ) ^{2}}

= - \frac{\frac{2 ^{2} e ^{4}}{( e ^{4} - 1 ) ^{2}} ( e ^{4} - 1 )}{e ^{4} + 1}

Multiply

\frac{2 ^{2} e ^{4}}{( e ^{4} - 1 ) ^{2}} (e^{4} - 1) : \frac{4 e ^{4}}{e ^{4} - 1}

\frac{2 ^{2} e ^{4}}{( e ^{4} - 1 ) ^{2}} (e^{4} - 1)

Multiply fractions:

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

= \frac{2 ^{2} e ^{4} ( e ^{4} - 1 )}{( e ^{4} - 1 ) ^{2}}

Cancel the common factor:

e^{4} - 1

= \frac{2 ^{2} e ^{4}}{e ^{4} - 1}

2^{2} = 4

= \frac{4 e ^{4}}{e ^{4} - 1}

= - \frac{\frac{4 e ^{4}}{e ^{4} - 1}}{e ^{4} + 1}

Apply the fraction rule:

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

= - \frac{4 e ^{4}}{( e ^{4} - 1 ) ( e ^{4} + 1 )}

= - \frac{4 e ^{4}}{( e ^{4} - 1 ) ( e ^{4} + 1 )}

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Frequently Asked Questions (FAQ)

What is the value of (-csch^2(2))/(coth(2)) ?
The value of (-csch^2(2))/(coth(2)) is -(4e^4)/((e^4-1)(e^4+1))