Question

Prove the following trigonometric identities.

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

Answer 1

We have to prove `tan A/(1 + tan^2  A)^2 + cot A/((1 + cot^2 A)) = sin A  cos A`

We know that `sin^2 A + cos^2 A = 1`

So

`tan A/(1 + tan^2 A)^2 + cot A/(1 + cot^2 A)^2`

`= tan A/(sec^2 A)^2  + cot A/(cosec^2 A)^2`

`= tan A/sec^4 A  + cot A/(cosec^4 A)`

`= (sin A/cos A)/(1/cos^4 A) + (cos A/sin A)/(1/sin^4 A)`

`= (sin A cos^4 A)/cos A + (cos A sin^4 A)/sin A`

`= sin A cos^3 A + cos A sin^3 A`

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

= sin A cos A

Hence proved.