To prove the given equation:

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

We know the identity:

sin^2x + cos^2x = 1

So, cos^2x = 1 - sin^2x

Substitute this into the given equation:

cos x + (1 - sin^2x) = 1/2 - sin^2x

Rearrange terms:

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

cos x + 1 = 1/2

cos x = -1/2

Therefore, the given equation is not correct.