We can define angles between subspaces using principal angles. For subspace spanned by orthonormal columns of matrix $U_{n\times r}$ and $X_{n\times r}$ we can define $\cos \theta \,=\, \sigma_{\min}\left(U^TX\right)$, where $\sigma_{\min}\left(A\right)$ denotes the minimum singular value of matrix $A$. Similarly, $\sin \theta \,=\, \sigma_{\max}\left(V^TX\right)$, where $V_{n \times n-r}$ is subspace complementary to $U$ i.e. $UU^T + VV^T = I_n$ and $U^TV = 0$.
How can I show: $\sin ^2\theta + \cos^2 \theta = 1$?