Answer 1

We have,

tan y dx + tan x dy = 0

\[\Rightarrow \tan x\frac{dy}{dx} = - \tan y \]

\[ \Rightarrow \cot y dy = - \cot x dx\]

Integrating both sides, we get

\[\int\cot y dy = - \int\cot x dx\]

\[ \Rightarrow \log \left| \sin y \right| = - \log \left| \sin x \right| + \log C\]

\[ \Rightarrow \log \left| \sin y \right| + \log \left| \sin x \right| = \log C\]

\[ \Rightarrow \log \left| \left( \sin y \right)\left( \sin x \right) \right| = \log C\]

\[ \Rightarrow \left( \sin y \right)\left( \sin x \right) = C\]

\[ \Rightarrow \sin x \sin y = C\]