Prove the identify on the left is equal to the right sin x

Prove the identify on the left is equal to the right (sin x - cos x) (sin x + cos x^) = - cos x

Solution

(sinx-cosx)(sinx+cosx) = -cosx

the identity you have proved is wrong

you can verify it by substituting x = pi/4

sin(pi/4) = 1/sqrt(2)

cos(pi/4) = 1/sqrt(2)

sinx - cosx = 0

which makes lhs = 0

rhs = -cosx = -1/sqrt(2)

so the identity is wrong.