Let $u(x, y)$ and $v(x, y)$ be a pair of conjugate harmonic functions in a domain $D$.

Prove that

$$U(x, y)=e^{-2 u v} \cos \left(u^{2}-v^{2}\right) \quad \text { and } \quad V(x, y)=e^{-2 u v} \sin \left(u^{2}-v^{2}\right)$$

also form a pair of conjugate harmonic functions in $D$.