\begin{align*}
0&=\text{Cov}(U,V)\\
&=\text{Cov}(X\cos(\theta)+Y\sin(\theta),Y\cos(\theta)-X\sin(\theta))\\
&=\text{Cov}(X\cos(\theta), Y\cos(\theta)-X\sin(\theta))+\text{Cov}(Y\sin(\theta), Y\cos(\theta)-X\sin(\theta))\\
&=\text{Cov}(X\cos(\theta), Y\cos(\theta))-\text{Cov}(X\cos(\theta),X\sin(\theta))+\text{Cov}(Y\sin(\theta), Y\cos(\theta))\\
&\;\;\;-\text{Cov}(Y\sin(\theta),X\sin(\theta))\\
&=\cos^2(\theta)\text{Cov}(X, Y)-\cos(\theta)\sin(\theta)\text{Cov}(X,X)+\sin(\theta)\cos(\theta)\text{Cov}(Y, Y)-\sin^2(\theta)\text{Cov}(Y,X)\\