1. **State the problem:** Prove that $$\cos^4 x - \sin^4 x = \cos 2x$$. 2. **Recall the formula:** The difference of squares formula states $$a^2 - b^2 = (a-b)(a+b)$$. 3. **Apply the difference of squares:** $$\cos^4 x - \sin^4 x = (\cos^2 x)^2 - (\sin^2 x)^2 = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)$$ 4. **Use the Pythagorean identity:** $$\cos^2 x + \sin^2 x = 1$$ 5. **Simplify:** $$ (\cos^2 x - \sin^2 x) \times 1 = \cos^2 x - \sin^2 x $$ 6. **Recall the double angle formula for cosine:** $$ \cos 2x = \cos^2 x - \sin^2 x $$ 7. **Conclusion:** $$ \cos^4 x - \sin^4 x = \cos 2x $$ This completes the proof.