The Solution of differential equation $\cos x\sin ydx + \sin x\cos ydy = 0$ is

Given differential equation is

$\cos x\sin ydx + \sin x\cos ydy = 0$

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

$\Rightarrow$ $\frac{{\cos x}}{{\sin x}}dx = - \frac{{\cos y}}{{\sin y}}dy$

$\Rightarrow$ $\cot xdx = - \cot ydy$

On integrating both sides,

we get
$\log \sin x = - \log \sin y + \log C$

$\Rightarrow$ $\log \sin x\sin y = \log C$

$\Rightarrow$ $\sin x \cdot \sin y = C$