Use the fundamental identities and algebraic techniques to p

Use the fundamental identities and algebraic techniques to prove the following identities.

Use the fundamental identities and algebraic techniques to prove the following identities. tan^2 x - cot^2 x = sec^2 x - csc^2 x

Solution

tan^2 x - cot^2 x = sec^2 x - csc^2 x

LHS = tan^2 x - cot^2 x = (sin^2 x/cos^2 x) - (cos^2 x/sin^2 x)

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

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

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

= 1/cos^2 x - 1/sin^2 x

= sec^2 x - csc^2 x = RHS