  We use the geometric concept of principal angles between subspaces to compute
the noncommutative distribution of an expression involving two free
projections. For example, this allows to simplify a formula by
Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We
also derive economically an explicit formula for the free additive convolution
of Bernoulli distributions. As a byproduct, we observe the remarkable fact that
the principal angles between random half-dimensional subspaces are
asymptotically distributed according to the uniform measure.
