prove the identity :

Question

cos^4 theta- sin ^4 thetaover sin squared theta cos squared theta =cot squared theta - tan squared theta

Answer 1

I'll use x for theta, for my convenience.

(cos^4x - sin^4x)/sin^2x cos^2x

= (cos^2x - sin^2x)(cos^2x + sin^2x)/sin^2x cos^2x

= (cos^2x - sin^2x)/sin^2x * (cos^2x + sin^2x)/cos^2x

= (cot^2x - 1)(1+tan^2x)
= cot^2x - 1 + cot^2xtan^2x - tan^2x
= cot^2x - 1 + 1 - tan^2x
= cot^2x - tan^2x