Here is a simple proof for $k=2$. Karl's examples show that the statement is false in general (without more careful assumptions).

**Claim** If $\mathscr I, \mathscr J\subseteq \mathscr O_X$ are two ideal sheaves such that $\mathscr I +\mathscr J =\mathscr O_X$, then $\mathscr I\mathscr J=\mathscr I\cap \mathscr J$.

**Proof** 
$$
\qquad \qquad\qquad \mathscr{(I+J)(I\cap J)}\subseteq \mathscr{I J}\subseteq \mathscr{I\cap J}.  \qquad\qquad\qquad \square
$$