If `tan^(-1)((x^2-y^2)/(x^2+y^2))=a` , prove that `(dy)/(dx)=x/y((1-tana))/((1+tana))` .

Explanation

Given `\tan ^{-1}((x^{2}-y^{2}) /(x^{2}+y^{2}))=a`
$$
\begin{aligned}
&\Rightarrow\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\frac{\tan a}{1} \\
&\Rightarrow\left(\frac{x^{2}-y^{2}+x^{2}+y^{2}}{x^{2}-y^{2}-x^{2}-y^{2}}\right)=\frac{\tan a+1}{\tan a-1} \\
&\Rightarrow \quad-\frac{x^{2}}{y^{2}}=\frac{\tan a+1}{\tan a-1} \\
&\Rightarrow \quad \frac{y^{2}}{x^{2}}=\frac{1-\tan a}{\tan a+1} \\
&\Rightarrow \quad y^{2}=\left(\frac{1-\tan a}{\tan a+1}\right) x^{2} \\
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