Trigonometry · Challenge problem

Vectors $\mathbf u$ and $\mathbf v$ have $|\mathbf u| = 3$ and $|\mathbf v| = 2$, and $\mathbf u + 2\mathbf v$ is perpendicular to $\mathbf u - \mathbf v$. Find the cosine of the angle between $\mathbf u$ and $\mathbf v$.