Question

sec^6 theta - tan^6 theta = 1+3 sec^2theta ×tan^2theta​

Answer 1

Step-by-step explanation:

Given :

sec⁶θ - tan⁶θ = 1 + 3 sec²θ tan²θ

We can also write this as :

sec⁶θ = tan⁶θ + 1 + 3 sec²θ tan²θ

Now,

LHS = sec⁶θ

= (sec²θ)³

[sec²θ = 1 + tan²θ]

= (1 + tan²θ)³

[(a + b)³ = a³ + b³ + 3ab(a + b)]

Here, a = 1, b = tan²θ

= 1 + tan⁶θ + 3tan²θ(1 + tan²θ)

= 1 + tan⁶θ + 3tan²θ(sec²θ)

= 1 + tan⁶θ + 3tan²θ sec²θ

= RHS

Hence proved !!