What is the general solution for (1-tan^2(A))/(2tan(A))=cot(A) ?

The general solution for (1-tan^2(A))/(2tan(A))=cot(A) is No Solution for A\in\mathbb{R}

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(1-tan^2(A))/(2tan(A))=cot(A)

Solution

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} = cot (A)

Solution

No Solution for A \in R

Solution steps

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} = cot (A)

Subtract

cot (A)

from both sides

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A) = 0

Simplify

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A) : \frac{1 - tan ^{2} ( A ) - 2 cot ( A ) tan ( A )}{2 tan ( A )}

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A)

Convert element to fraction:

cot (A) = \frac{cot ( A ) 2 tan ( A )}{2 tan ( A )}

= \frac{1 - tan ^{2} ( A )}{2 tan ( A )} - \frac{cot ( A ) \cdot 2 tan ( A )}{2 tan ( A )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - tan ^{2} ( A ) - cot ( A ) \cdot 2 tan ( A )}{2 tan ( A )}

\frac{1 - tan ^{2} ( A ) - 2 cot ( A ) tan ( A )}{2 tan ( A )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 - tan^{2} (A) - 2 cot (A) tan (A) = 0

Rewrite using trig identities

1 - tan^{2} (A) - 2 cot (A) tan (A)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= 1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )}

Simplify

1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )} : - \frac{1}{cot ^{2} ( A )} - 1

1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )}

(\frac{1}{cot ( A )})^{2} = \frac{1}{cot ^{2} ( A )}

(\frac{1}{cot ( A )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cot ^{2} ( A )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( A )}

2 cot (A) \frac{1}{cot ( A )} = 2

2 cot (A) \frac{1}{cot ( A )}

Multiply fractions:

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

= \frac{1 \cdot 2 cot ( A )}{cot ( A )}

Cancel the common factor:

cot (A)

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= 1 - \frac{1}{cot ^{2} ( A )} - 2

Group like terms

= - \frac{1}{cot ^{2} ( A )} + 1 - 2

Add/Subtract the numbers:

1 - 2 = - 1

= - \frac{1}{cot ^{2} ( A )} - 1

= - \frac{1}{cot ^{2} ( A )} - 1

- 1 - \frac{1}{cot ^{2} ( A )} = 0

Solve by substitution

- 1 - \frac{1}{cot ^{2} ( A )} = 0

Let:

cot (A) = u

- 1 - \frac{1}{u ^{2}} = 0

- 1 - \frac{1}{u ^{2}} = 0 : u = i, u = - i

- 1 - \frac{1}{u ^{2}} = 0

Multiply both sides by

u^{2}

- 1 - \frac{1}{u ^{2}} = 0

Multiply both sides by

u^{2}

- 1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

- 1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

- 1 \cdot u^{2} : - u^{2}

- 1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= - u^{2}

Simplify

- \frac{1}{u ^{2}} u^{2} : - 1

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

Multiply fractions:

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

= - \frac{1 \cdot u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= - 1

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- u^{2} - 1 = 0

- u^{2} - 1 = 0

- u^{2} - 1 = 0

Solve

- u^{2} - 1 = 0 : u = i, u = - i

- u^{2} - 1 = 0

Move

1

to the right side

- u^{2} - 1 = 0

Add

1

to both sides

- u^{2} - 1 + 1 = 0 + 1

Simplify

- u^{2} = 1

- u^{2} = 1

Divide both sides by

- 1

- u^{2} = 1

Divide both sides by

- 1

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

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

u = i, u = - i

Substitute back

u = cot (A)

cot (A) = i, cot (A) = - i

cot (A) = i, cot (A) = - i

cot (A) = i :

No Solution

cot (A) = i

No Solution

cot (A) = - i :

No Solution

cot (A) = - i

No Solution

Combine all the solutions

No Solution for A \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (1-tan^2(A))/(2tan(A))=cot(A) ?
The general solution for (1-tan^2(A))/(2tan(A))=cot(A) is No Solution for A\in\mathbb{R}