Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$? I am finishing a proof.  It seems like I can use $\cos^2 + \sin^2 = 1$ to figure this out, but I just can't see how it works.  So I've got two questions.  
Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$? 
And if it does, then how?

Not really different but just another way of looking at it. Start with $\sin^2x+\cos^2x=1$ and subtract $2\cos^2x$ from both sides. Done!
