Two sources of sound S_1 and S_2 vibrate at the same frequency and...

From the given figure, point $P$ lies on the perpendicular bisector of $S_1S_2$, so the path difference is zero ($\Delta x = 0$).

Let the intensity of each individual source at $P$ be $I'$.

Since the sources are in phase and the path difference is zero, they interfere constructively. The resultant intensity is:

$I_0 = I' + I' + 2\sqrt{I' I'} \cos 0^\circ = 4I'$

If one source is switched off, the intensity at $P$ is solely due to the single remaining source, which is $I'$.

New intensity $= I' = \dfrac{I_0}{4}$

This result depends only on the path difference being zero and is completely independent of the angle $\theta$.