Verify the identity $\cos^2x-\sin^2x = 2\cos^2x-1$ I am having problems understanding how to verify this identity. I am quite sure that it is to be solved using the Pythagorean identities but, alas, I'm not seeing what might otherwise be obvious. 
I need to verify the identity 
$$\cos^2x-\sin^2x = 2\cos^2x-1$$
Thank you for your help.

Yes, indeed, we can use the Pythagorean Identity:
$$\cos^2 x + \sin^2 x = 1 \iff \color{blue}{\sin^2 x = 1-\cos^2 x}$$

$$\begin{align}\cos^2x-\color{blue}{\sin^2x} & = \cos^2 x - \color{blue}{(1-\cos^2 x)}\\\\
&= \cos^2 x - 1 + \cos^2 x \\ \\ 
& = 2\cos^2 x - 1\end{align}$$
