Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
sec⁶ A - tan⁶ A = 1 + 3 tan² A + 3 tan⁴ A.

126 Likes

a³ - b³ = (a - b)³ + 3ab(a - b)

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

⇒ sec⁶ A - tan⁶ A = (sec² A - tan² A)³ + 3sec² A tan² A(sec² A - tan² A)

⇒ 1³ + 3sec² A tan² A × 1

⇒ 1 + 3sec² A tan² A

⇒ 1 + 3(1 + tan² A)tan² A

⇒ 1 + 3(tan⁴ A + tan² A)

⇒ 1 + 3 tan² A + 3 tan⁴ A

Since, L.H.S. = R.H.S. hence, proved that sec⁶ A - tan⁶ A = 1 + 3tan² A + 3tan⁴ A.

Answered By

66 Likes