Question

Prove the following identities

(i) sec⁶ θ = tan⁶ θ + 3tan² θ sec² θ + 1

(ii) (sinθ + secθ)² + (cosθ + cosecθ)² = 1 + (secθ + cosecθ)²

Answer 1

(i) L.H.S = sec⁶ θ = (sec² θ)³ = (1 + tan² θ)³ = (tan² θ + 1)³ 

(a + b)³ = a³ + 3a² b + 3ab² + b³ 

= (tan² θ)³ + 3(tan² θ)² x 1 + 3 x tan² θ x 1² + 1 

= tan⁶ θ + 3tan² θ x (sec² θ – 1) + 3tan² θ + 1 

= tan⁶ θ + 3tan² θsec² θ – 3tan² θ + 3tan² θ + 1 

= tan⁶ θ + 3tan² θ sec² θ + 1 = R.H.S

(ii) L.H.S = (sinθ + secθ)² + (cosθ + cosecθ)²

= sin² θ + 2sinθ secθ + sec² θ + cos² θ + 2cosθ cosecθ + cosec² θ