LHS $= \cos\left(\frac{\pi}{4} - x\right) \cos\left(\frac{\pi}{4} - y\right) - \sin\left(\frac{\pi}{4} - x\right) \sin\left(\frac{\pi}{4} - y\right)$

We know that $\cos(A + B) = \cos A \cos B - \sin A \sin B$.

Let $A = \left(\frac{\pi}{4} - x\right)$ and $B = \left(\frac{\pi}{4} - y\right)$.

Therefore,

$$\text{LHS} = \cos\left[\left(\frac{\pi}{4} - x\right) + \left(\frac{\pi}{4} - y\right)\right]$$ $$= \cos\left[\frac{\pi}{4} - x + \frac{\pi}{4} - y\right]$$ $$= \cos\left[\frac{\pi}{2} - (x + y)\right]$$

Using the identity $\cos\left(\frac{\pi}{2} - x\right) = \sin x$, we get:

$$= \sin(x + y)$$ $$= \textbf{RHS}$$

Hence proved.