1. **State the problem:** We want to add the basic trigonometric identity for $\cos 2a$ and then simplify the result. 2. **Recall the identity:** The double-angle formula for cosine is: $$\cos 2a = \cos^2 a - \sin^2 a$$ 3. **Add the identity to itself:** Adding $\cos 2a$ to $\cos 2a$ means: $$\cos 2a + \cos 2a = 2 \cos 2a$$ 4. **Substitute the identity:** Replace $\cos 2a$ with $\cos^2 a - \sin^2 a$: $$2(\cos^2 a - \sin^2 a) = 2\cos^2 a - 2\sin^2 a$$ 5. **Simplify:** This is the simplified form of the sum: $$2\cos^2 a - 2\sin^2 a$$ **Final answer:** $$\cos 2a + \cos 2a = 2\cos 2a = 2\cos^2 a - 2\sin^2 a$$