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Question
Prove the following trigonometric identities.
tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B
Solution
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
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