Question

Let $f(x)=\cos \left(2{\tan }^{-1}\sin \left({\cot }^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$, $0<x<1$. Then:

• A. $(1-x{)}^2{f}^{\prime }(x)-2(f(x){)}^2=0$
• B. $(1+x{)}^2{f}^{\prime }(x)+2(f(x){)}^2=0$
• C. $(1-x{)}^2{f}^{\prime }(x)+2(f(x){)}^2=0$
• D. $(1+x{)}^2{f}^{\prime }(x)-2(f(x){)}^2=0$

Answer 1

Answer: (C)

$f(x)=\cos \left(2{\tan }^{-1}\sin \left({\cot }^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$
${\cot }^{-1}\sqrt{\frac{1-x}{x}}={\sin }^{-1}\sqrt{x}$
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 $f^{\prime }(x)=\frac{-2}{(1+x{)}^2}$
or $f^{\prime }(x)(1-x{)}^2=-2{\left(\frac{1-x}{1+x}\right)}^2$
or $(1-x{)}^2{f}^{\prime }(x)+2(f(x){)}^2=0$