Let f(x)=cos

$f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$

$\cot ^{-1} \sqrt{\frac{1-x}{x}}=\sin ^{-1} \sqrt{x}$

or $f(x)=\cos \left(2 \tan ^{-1} \sqrt{x}\right)$

$=\cos \tan ^{-1}\left(\frac{2 \sqrt{x}}{1-x}\right)$

$f(x)=\frac{1-x}{1+x}$

Now $\mathrm{f}^{\prime}(\mathrm{x})=\frac{-2}{\left(1+\mathrm{x}^{2}\right)}$

or $f^{\prime}(x)(1-x)^{2}=-2\left(\frac{1-x}{1+x}\right)^{2}$

or $(1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$