Question

Prove the following trigonometric identities.

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

Answer 1

We have to prove  `(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`

We know that, `sin^2 theta + cos^2 theta = 1`

`(1 + sec theta)/sec theta = (1 + 1/cos theta)/(1/cos theta)`

`= ((cos theta + 1)/cos theta)/(1/cos theta)`

`= (1 + cos theta)/1`

Multiplying the numerator and denominator by `(1 - cos theta)` we have

`(1 + sec theta)/sec theta  = ((1 + cos theta)(1 - cos theta))/(1- cos theta)`

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

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