The solution of the differential equation $(\cot y) dx=x ~dy$ is :

$x=C \cos y$ $x=C \sec y$ $y=C \cos x$ $y=C \sec x$

$x=C \sec y$

cot y dx = x dy so  $\frac{1}{x} dx = \frac{1}{\cot y}dy$ ⇒  $\int \frac{1}{x}dx = \int \tan y dy$ integrating both sides w.r.t x ⇒  $\log x = \log \sec y + \log k$ taking constant in terms of log for case of calculation so $\log x = \log k \sec y$ ⇒  $x = k \sec y$ let arbitrary constant k = C x = C sec y