This question tests the application of cosine addition and subtraction formulas. The key insight is that when we subtract $\cos(A-B)$ from $\cos(A+B)$, the $\cos A \cos B$ terms cancel out, leaving only $-2 \sin A \sin B$.

Students must also recall that $\sin\left(\frac{3\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ since $\frac{3\pi}{4}$ lies in the second quadrant where sine is positive.

This follows the same pattern as Example 15 in the textbook which proves $\cos\left(\frac{\pi}{4} + x\right) + \cos\left(\frac{\pi}{4} - x\right) = \sqrt{2}\,\cos x$.