Question

Prove that `1/(1 + tan^2 A) + 1/(1 + cot^2 A) = 1`.

Answer 1

Given: A angle A (any real angle)

To Prove: `1/(1 + tan^2 A) + 1/(1 + cot^2 A) = 1`

Proof [Step-wise]:

1. Write tan and cot in terms of sin and cos:

`tan A = sin A/cos A`

`cot A = cos A/sin A`

2. Evaluate the first term:

`1/(1 + tan^2 A) = 1/(1 + (sin^2 A/cos^2 A))`

= `1/((cos^2 A  +  sin^2 A)/cos^2 A)`

= `(cos^2 A)/(sin^2 A + cos^2 A)`

= cos² A

3. Evaluate the second term similarly:

`1/(1 + cot^2 A) = 1/(1 + (cos^2 A/sin^2 A))`

= `1/((sin^2 A  +  cos^2 A)/sin^2 A)`

= `(sin^2 A)/(sin^2 A + cos^2 A)` 

= sin² A

4. Add the two results:

`1/(1 + tan^2 A) + 1/(1 + cot^2 A) = cos^2 A + sin^2 A = 1`, using the Pythagorean identity sin² A + cos² A = 1.

Hence, `1/(1 + tan^2 A) + 1/(1 + cot^2 A) = 1`, as required.