If `sec((x+y)/(x-y))=a` , prove that `(dy)/(dx)=y/x` .

Explanation

\begin{equation}
\begin{aligned}
&\text { Given } \sec ((x+y) /(x-y))=a\\
&\Rightarrow\left(\frac{x+y}{x-y}\right)=\frac{\sec ^{-1}(a)}{1}\\
&\Rightarrow\left(\frac{x+y+x-y}{x+y-x+y}\right)=\frac{\sec ^{-1}(a)+1}{\sec ^{-1}(a)-1}\\
&\Rightarrow \frac{x}{y}=\frac{\sec ^{-1}(a)+1}{\sec ^{-1}(a)-1}\\
&\Rightarrow \frac{y}{x}=\frac{\sec ^{-1}(a)-1}{\sec ^{-1}(a)+1}\\
&\Rightarrow y=\left(\frac{\sec ^{-1}(a)-1}{\sec ^{-1}(a)+1}\right) x\\
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