Question

Prove the identity: (1 + tan α tan β)² + (tan α – tan β)² = sec² α sec² β

Answer 1

Let us consider the LHS

(1 + tan α tan β)² + (tan α – tan β)²

1+ tan² α tan² β + 2 tan α tan β + tan² α + tan² β – 2 tan α tan β

1 + tan² α tan² β + tan² α + tan² β

tan² α (tan² β + 1) + 1 (1 + tan² β)

(1 + tan² β) (1 + tan² α)

As we know, 1 + tan² θ = sec² θ

Therefore,

sec² α sec² β

= RHS

∴ LHS = RHS

Thus proved.