Determine if the following is an identity: $$\frac{\cos 2\theta}{\sin \theta + \cos \theta} = \cos \theta - \sin \theta$$

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

$$\begin{array}{rclc} \frac{\cos 2\theta}{\sin \theta + \cos \theta} &=& \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta + \cos \theta} & \overset{\normalsize{\textrm{rewriting things so that all trig functions are of a}}}{\scriptsize{\textrm{single angle, by recalling } \cos 2\theta = \cos^2 \theta - \sin^2 \theta}}\\\\ &=& \frac{(\cos \theta + \sin \theta)(\cos \theta - \sin \theta)}{\cos \theta - \sin \theta} & \overset{\normalsize{\textrm{trying to eliminate the denominator,}}}{\scriptsize{\textrm{we factor, and hope something cancels}}}\\\\ &=& (\cos \theta + \sin \theta) \cdot \cancel{\frac{\cos \theta - \sin \theta}{\cos \theta - \sin \theta}} & \scriptsize{\textrm{...which it does!}}\\\\ &=& \cos \theta + \sin \theta & \end{array}$$

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