Question

Prove the following trigonometric identities.

`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`

Answer 1

In the given question, we need to prove `(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`

Using the identity `a^2 - b^2  = (a + b)(a - b)`

`cos theta/((cosec theta + 1)) + cos theta/(cosec theta - 1) = (cos theta(cosec theta - 1)+ cos theta(cosec theta + 1))/(cosec^2 theta - 1)`

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

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

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

`= 2 sin theta/cos theta`

`= 2 tan theta`

Hence proved.