How can I prove that a specific symmetric matrix $P = I - \frac{uu^T}{u^T u}$ is idempotent?

Question

Given the matrix $$P = I - \frac{uu^T}{u^T u},$$ how can I prove that $P = P^2$?

I know that the matrix is symmetric, but how can I use this information to prove that it's idempotent?