Question

Prove the following trigonometric identities.

(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)

Answer 1

We have to prove

(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)

Consider the LHS.

`(cosec θ − sec θ) (cot θ − tan θ) = (1/sin theta - 1/cos theta)(cos theta/sin theta - sin theta/cos theta)`

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

`= (cos theta - sin theta)/(sin theta cos theta) ((cos theta + sin theta)(cos theta - sin theta))/(sin theta cos theta)`

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

Now, consider the RHS.

`(cosec θ + sec θ) ( sec θ cosec θ − 2) = (1/sin theta + 1/cos theta) (1/cos theta 1/sin theta - 2)`

`= ((cos theta + sin theta)/(sin theta cos theta))((1- 2sin theta cos theta)/(sin theta cos theta))`

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

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

∴ LHS = RHS

Hence proved.