1. The problem is to simplify the expression $$\frac{\cos^2 a - \sin^2 a}{\cos^2 a (2 - \cos^2 a)}$$. 2. Recall the Pythagorean identity: $$\sin^2 a + \cos^2 a = 1$$. 3. Using this identity, rewrite the numerator: $$\cos^2 a - \sin^2 a = \cos^2 a - (1 - \cos^2 a) = \cos^2 a - 1 + \cos^2 a = 2\cos^2 a - 1$$. 4. Substitute back into the expression: $$\frac{2\cos^2 a - 1}{\cos^2 a (2 - \cos^2 a)}$$. 5. Factor the denominator: $$\cos^2 a (2 - \cos^2 a)$$ remains as is. 6. The expression is: $$\frac{2\cos^2 a - 1}{\cos^2 a (2 - \cos^2 a)}$$. 7. No common factors to cancel, so this is the simplified form. Final answer: $$\frac{2\cos^2 a - 1}{\cos^2 a (2 - \cos^2 a)}$$