Verify the identity using trig rules sin2 tan cos z 1 sin a

Verify the identity using trig rules. sin2() tan( cos (z) 1 sin (a sin2 (x) + tan2 (z) cos2 (x) + 1

Solution

(sin²x - tan²x)/(sin²x + tan²x) = (cos²x - 1)/(cos²x + 1)

LHS = (sin²x - tan²x)/(sin²x + tan²x) = (sin²x - sin²x/cos²x)/(sin²x + sin²x/cos²x)

= sin²x*(cos²x - 1)/sin²x*(cos²x + 1) = (cos²x - 1)/(cos²x + 1) = RHS