1. **State the problem:** Simplify the expression $$\frac{\cos^2 a - \sin^2 a}{\cos^2 a (2 - \cos^2 a)}$$. 2. **Recall the formula:** Use the Pythagorean identity and double-angle formulas: - $\cos^2 a - \sin^2 a = \cos 2a$ - $\sin^2 a = 1 - \cos^2 a$ 3. **Rewrite the denominator:** $$2 - \cos^2 a = 2 - \cos^2 a$$ (leave as is for now). 4. **Substitute numerator:** $$\frac{\cos 2a}{\cos^2 a (2 - \cos^2 a)}$$ 5. **No further factorization simplifies the denominator easily, so this is the simplified form.** **Final answer:** $$\boxed{\frac{\cos 2a}{\cos^2 a (2 - \cos^2 a)}}$$