We need to show that the product of orthogonal projection is orthogonal iff the projections commute.
$\Rightarrow$ Let $P, Q$ be the orthogonal projections, we know $P^2 = P$ and $Q^2 =Q$. Also, $PQ$ is a projection $(PQ)(PQ) = PQ = P^2Q^2$. From here, how do I go about showing the projections commute.(I intially though we can take inverse, but as mentioned in the comments the projections can be singular).
$\Leftarrow$ We can show that PQ is a projection given P and Q commute but how do I show it is orthogonal.
Note that the following statements are equivalent:
$R$ is an orthogonal projection
$R^2=R=R^*$
$R^*R=R$
Then
If $PQ=QP$, then $(PQ)^*PQ=Q^*P^*PQ=QPQ=PQ^2=PQ$.
Conversely, if $PQ$ is an orthogonal projection, we have $QP=Q^*P^*=(PQ)^*=PQ$.