Prove the identity:

Question

1+tan^2(u)/1-tan^2(u)= 1/ cos^2(u)- sin^2(u)?

Answer 1

The trigonometric equation is (1 + tan² u)/(1 - tan² u) = 1/(cos² u - sin² u).

Left hand side identity : (1 + tan² u)/(1 - tan² u).

Trigonometric function : tan x = sin x/cos x.

= (1 + (sin u/cos u)²) / (1 - (sin u/cos u)²)

= [(cos² u + sin² u)/cos² u] / [(cos² u - sin² u)/cos² u]

= (cos² u + sin² u) / (cos² u - sin² u)

Pythagorean identity : sin² x + cos² x = 1.

= 1 / (cos² u - sin² u)

= Right hand side identity.