Question

Prove the following trigonometric identities.

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

Answer 1

In the given question, we need to prove `1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Using `cot theta = cos theta/sin theta` and `cosec theta = 1/sin theta` We get

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

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

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

Further, using the property `sin^2 theta + cos^2 theta = 1`

We get

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

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

`= 1/sin theta`

`= cosec theta`

Hence proved.