Determine if the following is an identity: $$\cos 2\theta = 2\cos^2 \theta - 1$$

Let us start by trying to simplify the left side,$\require{cancel}$

$$\begin{array}{rclc} \cos 2\theta &=& \cos^2 \theta - \sin^2 \theta & \overset{\normalsize{\textrm{rewriting things so that all trig functions are of a}}}{\scriptsize{\textrm{single angle, by recalling } \cos 2\theta = \cos^2 \theta - \sin^2 \theta}}\\\\ &=& \cos^2 \theta - (1 - \cos^2 \theta) & \overset{\normalsize{\textrm{the right-hand side only involves cosines, so we "translate"}}}{\scriptsize{\sin^2 \theta \textrm{ into a cosine expression using } \cos^2 \theta + \sin^2 \theta = 1}}\\\\ &=& 2\cos^2 \theta - 1 \end{array}$$

Since this is identical to the right-hand side of the given equation, we can say that the above equation is an identity.