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

Question: If `f(x)=\cos \left[2 \tan ^{-1}\left(\sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right]\right]`

Options:

(a) `f^{\prime}(x)(x-1)^{2}-2(f(x))^{2}=0`

(b) `f^{\prime}(x)(x-1)^{2}+2(f(x))^{2}=0`

(c) `f^{\prime}(x)(x+1)^{2}+2(f(x))^{2}=0`

(d) `f^{\prime}(x)(x+1)^{2}-2(f(x))^{2}=0`

SOLUTION:

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

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

`=\left(\frac{2}{1+x}\right)-1 `

`\therefore f^{\prime}(x)=\frac{-2}{(1+x)^{2}}`

`\Rightarrow f^{\prime}(x)(1+x)^{2}=-2`