Question

Prove the identity: tan³ x/(1 + tan² x) + cot³ x/(1 + cot² x) = (1 - 2sin² x cos² x)/sin x cos x

Answer 1

Let us consider the LHS:

tan³ x/(1 + tan² x) + cot³ x/(1 + cot² x) 

On using the formula

1 + tan²x = sec²x and 1 + cot²x = cosec²x

As we know that, a² + b² = (a + b)² - 2ab

= RHS

∴ LHS = RHS

Thus proved.