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Question
tan y dx + tan x dy = 0
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Solution
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
\[\int\cot y dy = - \int\cot x dx\]
\[ \Rightarrow \log \left| \sin y \right| = - \log \left| \sin x \right| + \log C\]
\[ \Rightarrow \log \left| \sin y \right| + \log \left| \sin x \right| = \log C\]
\[ \Rightarrow \log \left| \left( \sin y \right)\left( \sin x \right) \right| = \log C\]
\[ \Rightarrow \left( \sin y \right)\left( \sin x \right) = C\]
\[ \Rightarrow \sin x \sin y = C\]
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