Question

$\frac{dy}{dx}+\frac{\cos x\sin y}{\cos y}=0$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}+\frac{\cos x\sin y}{\cos y}=0 \\ \Rightarrow \frac{dy}{dx}=-\frac{\cos x\sin y}{\cos y} \\ \Rightarrow \frac{\cos y}{\sin y}dy=-\cos xdx \\ \Rightarrow \cot ydy=-\cos xdx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \cot ydy=-\int \cos xdx \\ \Rightarrow \log \left|\sin y\right|=-\sin x+C \\ \mathrm{Hence},\log \left|\sin y\right|=-\sin x+C\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$