Question

Prove the following trigonometric identities.

sec A (1 − sin A) (sec A + tan A) = 1

Answer 1

We have to prove sec A(1 − sin A)(sec A + tan A) = 1

We know that sec² A − tan² A − 1 

So,

sec A(1 − sin A)(sec A + tan A) = {sec A(1 − sin A)}(sec A + tan A)

= (sec A − sec A sin A)(sec A + tan A)

= `(sec A - 1/cos A  sin A) (sec A + tan A)`     ...`(∵ sec theta = 1/costheta)`

= `(sec A - sin A/cos A) (sec A + tan A)`    ...`(∵ tan theta = sin theta/costheta)`

= (sec A − tan A)(sec A + tan A)

= sec² A − tan² A

= 1 = R.H.S.    ... (∵ sec² θ = 1 tan² θ)