$\sin x \sin y \, dx + \cos x \cos y \, dy = 0$
$\dfrac{\sin x \sin y \, dx}{\sin y \cos x} + \dfrac{\cos x \cos y \, dy}{\sin y \cos x} = 0$
$\dfrac{\sin x \, dx}{\cos x} + \dfrac{\cos y \, dy}{\sin y} = 0$
$\displaystyle -\int \dfrac{-\sin x \, dx}{\cos x} + \int \dfrac{\cos y \, dy}{\sin y} = 0$
$-\ln (\cos x) + \ln (\sin y) = \ln c$
$\ln \dfrac{\sin y}{\cos x}= \ln c$
$\dfrac{\sin y}{\cos x}= c$
$\sin y = c \cos x$ answer