# How do you use the trig identity cos2x = cos^2 x - sin^2 x  to verify that cos 2x = 2 cos^2 x -1?

From the identity ${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$ we can subtract ${\cos}^{2} \left(x\right)$ to obtain ${\sin}^{2} \left(x\right) = 1 - {\cos}^{2} \left(x\right)$. Then,
$\cos \left(2 x\right) = {\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)$
$= {\cos}^{2} \left(x\right) - \left(1 - {\cos}^{2} \left(x\right)\right)$
$= 2 {\cos}^{2} \left(x\right) - 1$