The solution of differential equation cos x sin y dx + sin x cos y dy…

The correct answer is Option (2) → $\sin x \sin y = C$ ##

Given differential equation is

$\cos x \sin y \, dx + \sin x \cos y \, dy = 0$

$\Rightarrow \cos x \sin y \, dx = -\sin x \cos y \, dy$

$\Rightarrow \frac{\cos x}{\sin x} \, dx = -\frac{\cos y}{\sin y} \, dy \quad \text{[applying variable separable method]}$

$\Rightarrow \cot x \, dx = -\cot y \, dy$

On integrating both sides, we get

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

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

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

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