The solution of the differential equation $cos x\, sin y \, dx +sinx \, cos ydy =0$ is :

sinx sin y = C, where C is a constant

The correct answer is Option (2) → $\sin x \sin y = C$, where C is a constant

$\cos x\sin y dx +\sin x \cos ydy =0$

$\cos x\sin ydx=-\sin x \cos ydy$

$\int-\cot xdx=\int\cot y dy$

so $-\log\sin y=\log\sin x+\log C$

$⇒\sin x \sin y = C$