Question:
$\operatorname{Let} f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$
$0
Correct Option: , 3
Solution:
$f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$
$\cot ^{-1} \sqrt{\frac{1-x}{x}}=\sin ^{-1} \sqrt{x}$
or $\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sqrt{\mathrm{x}}\right)$
$=\cos \tan ^{-1}\left(\frac{2 \sqrt{x}}{1-x}\right)$
$f(x)=\frac{1-x}{1+x}$
Now $f^{\prime}(x)=\frac{-2}{(1+x)^{2}}$
or $f^{\prime}(x)(1-x)^{2}=-2\left(\frac{1-x}{1+x}\right)^{2}$
or $(1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.