Question

Prove the following trigonometric identities.

(secθ + cosθ) (secθ − cosθ) = tan²θ + sin²θ

Answer 1

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

We know that

`sin^2 theta + cos^2 theta = 1`

`sec^2 theta - tan^2 theta = 1`

`(sec theta + cos theta)(sec theta - cos theta) = sec^2 theta - cos^2 theta`

`= (1 + tan^2 theta) - (1 - sin^2 theta)`

`= 1 + tan^2 theta - 1 + sin^2 theta`

`= tan^2 theta + sin^2 theta`