<s>[INST] 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?
 [/INST] Observe that
$$
\begin{align*}
\sin^2(x)-\cos^2(x)&=\sin^2(x)+\bigl(\cos^2(x)-\cos^2(x)\bigr)-\cos^2(x)\\
&= (\sin^2(x)+\cos^2(x))-2\cos^2(x)\\
&= 1-2\cos^2(x).
\end{align*}
$$
More easily, just subtract $2\cos^2(x)$ from both sides of $\sin^2(x)+\cos^2(x)=1$ to get the result.
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