Question

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

Answer 1

We have,
\[\frac{dy}{dx} + \frac{\cos x \sin y}{\cos y} = 0\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{\cos x \sin y}{\cos y}\]
\[ \Rightarrow \frac{\cos y}{\sin y}dy = - \cos x dx\]
\[ \Rightarrow \cot y dy = - \cos x dx\]
Integrating both sides, we get
\[\int\cot y dy = - \int\cos x dx\]
\[ \Rightarrow \log \left| \sin y \right| = - \sin x + C\]
\[\text{ Hence,} \log \left| \sin y \right| = - \sin x +\text{ C is the required solution .} \]