1. The problem is to verify the identity: $$\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$$. 2. Recall the difference of squares formula: $$a^2 - b^2 = (a-b)(a+b)$$. 3. Apply this to the left side: $$\sin^4 x - \cos^4 x = (\sin^2 x)^2 - (\cos^2 x)^2 = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)$$. 4. Use the Pythagorean identity: $$\sin^2 x + \cos^2 x = 1$$. 5. Substitute into the expression: $$(\sin^2 x - \cos^2 x) \times 1 = \sin^2 x - \cos^2 x$$. 6. Rewrite the right side: $$1 - 2\cos^2 x$$. 7. Use the identity $$\sin^2 x = 1 - \cos^2 x$$ to rewrite the left side: $$\sin^2 x - \cos^2 x = (1 - \cos^2 x) - \cos^2 x = 1 - 2\cos^2 x$$. 8. Both sides are equal, so the identity is verified. Final answer: $$\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$$ is true.