Question

Prove the following trigonometric identities.

tan² A sec² B − sec² A tan² B = tan² A − tan² B

Answer 1

LHS = `tan^2 A sec^2 B - sec^2 A tan^2 B`

`= tan^2 A + (1 + tan^2 B) - sec^2 A (tan^2 A)`

`= tan^2 A + tan^2 A tan^2 B - tan^2 B(1 + tan^2 A)`     (`∵ sec^2 A = 4 tan^2 A`)

`= tan^2 A + tan^2 A tan^2 B - tan^2 B - tan^2 B tan^2 A`

`= tan^2 A - tan^2 B`

= RHS