Question

Prove the following trigonometric identities.

`((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`

Answer 1

We need to prove `((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`

Solving the L.H.S, we get

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

Using `sec theta = 1/cos theta, cot theta = cos theta/sin theta`. `cosec theta = 1/sin theta` we get

`(sec^2 theta(cot theta))/(cosec^2 theta) = (1/cos^2 theta (cos theta/sin theta))/(1/sin^2 theta)`

`= (1/(cos theta sin theta))/(1/sin^2 theta)`

`= sin^2 theta/(cos theta sin theta)`

`= sin theta/cos theta`

`= tan theta`

Hence proved.