Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

(1 + tan A)² + (1 - tan A)² = 2 sec² A.

The L.H.S of the equation can be written as,

⇒ 1 + tan² A + 2 tan A + 1 + tan² A - 2 tan A

⇒ 2(1 + tan² A)

⇒ 2 sec² A.

Since, L.H.S. = R.H.S hence, proved that (1 + tan A)² + (1 - tan A)² = 2 sec² A.