1. We need to prove the trigonometric identity: $$\frac{1-\sin^4\alpha}{\cos^2\alpha} - 2\sin^2\alpha = \cos^2\alpha$$. 2. Recall the Pythagorean identity: $$\sin^2\alpha + \cos^2\alpha = 1$$. 3. Start by factoring the numerator of the fraction: $$1 - \sin^4\alpha = (1 - \sin^2\alpha)(1 + \sin^2\alpha)$$. 4. Substitute this back into the expression: $$\frac{(1 - \sin^2\alpha)(1 + \sin^2\alpha)}{\cos^2\alpha} - 2\sin^2\alpha$$. 5. Using the Pythagorean identity, replace $$1 - \sin^2\alpha$$ with $$\cos^2\alpha$$: $$\frac{\cos^2\alpha (1 + \sin^2\alpha)}{\cos^2\alpha} - 2\sin^2\alpha$$. 6. Cancel $$\cos^2\alpha$$ in numerator and denominator: $$\frac{\cancel{\cos^2\alpha} (1 + \sin^2\alpha)}{\cancel{\cos^2\alpha}} - 2\sin^2\alpha = 1 + \sin^2\alpha - 2\sin^2\alpha$$. 7. Simplify the terms: $$1 + \sin^2\alpha - 2\sin^2\alpha = 1 - \sin^2\alpha$$. 8. Again, use the Pythagorean identity to replace $$1 - \sin^2\alpha$$ with $$\cos^2\alpha$$. 9. Therefore, the left side simplifies to $$\cos^2\alpha$$, which equals the right side. 10. Hence, the identity is proven: $$\frac{1-\sin^4\alpha}{\cos^2\alpha} - 2\sin^2\alpha = \cos^2\alpha$$.