1. **State the problem:** Simplify and verify the expression $$\frac{\cos^2 x - \sin^2 x}{\cos x - \sin x} = \cos x + \sin x.$$\n\n2. **Rewrite the numerator using a trigonometric identity:** Recall that $$\cos^2 x - \sin^2 x = \cos(2x).$$ So the expression becomes $$\frac{\cos(2x)}{\cos x - \sin x}.$$\n\n3. **Express the right side:** The right side is $$\cos x + \sin x.$$\n\n4. **Goal:** Show that $$\frac{\cos(2x)}{\cos x - \sin x} = \cos x + \sin x.$$\n\n5. **Multiply both sides by $$\cos x - \sin x$$ to check equality:**\n$$\cos(2x) = (\cos x + \sin x)(\cos x - \sin x).$$\n\n6. **Expand the right side:**\n$$ (\cos x + \sin x)(\cos x - \sin x) = \cos^2 x - \sin^2 x.$$\n\n7. **Recall the identity:** $$\cos^2 x - \sin^2 x = \cos(2x).$$\n\n8. **Therefore:**\n$$\cos(2x) = \cos(2x),$$ which confirms the equality.\n\n**Final answer:** $$\boxed{\frac{\cos^2 x - \sin^2 x}{\cos x - \sin x} = \cos x + \sin x}.$$