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.
$\endgroup$ 23 Answers
$\begingroup$Yes, indeed, we can use the Pythagorean Identity:
$$\cos^2 x + \sin^2 x = 1 \iff \color{blue}{\sin^2 x = 1-\cos^2 x}$$
$$\begin{align}\cos^2x-\color{blue}{\sin^2x} & = \cos^2 x - \color{blue}{(1-\cos^2 x)}\\\\ &= \cos^2 x - 1 + \cos^2 x \\ \\ & = 2\cos^2 x - 1\end{align}$$
$\endgroup$ $\begingroup$Remember that $sin^2(x) = 1 - cos^2(x)$.
Then $cos^2x - sin^2x = ?$
$\endgroup$ 0 $\begingroup$we have $\cos(x)^2-\sin(x)^2=\cos(x)^2-(1-\cos(x)^2)=2\cos(x)^2-1$ since $\sin(x)^2+\cos(x)^2=1$
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