Q104 / 180 · KCET 2011 · Mathematics

If $f(x)=\frac{\sin ^{2} x}{1+\cot x}+\frac{\cos ^{2} x}{1+\tan x}$, then $f^{\prime}\left(\frac{\pi}{4}\right)$ is

  • A
    $\sqrt{3}$
  • B
    $\frac{1}{\sqrt{3}}$
  • Correct answer
    C
    0
  • D
    $-\sqrt{3}$

Solution

Given, $f(x)=\frac{\sin ^{2} x}{1+\cot x}+\frac{\cos ^{2} x}{1+\tan x}$ $$ \begin{gathered} f(x)=\frac{\sin ^{2} x}{1+\frac{1}{\tan x}}+\frac{\cos ^{2} x}{1+\tan x} \\ f(x)=\frac{\tan x \cdot \sin ^{2} x+\cos ^{2} x}{(1+\tan x)}=\frac{\sin ^{3} x+\cos ^{3} x}{(\cos x+\sin x)} \\ f(x)=\frac{(\sin x+\cos x)}{(\sin x+\cos x)}\left(\sin ^{2} x+\cos ^{2} x\right. \\ f^{\prime}\left(\frac{\pi}{4}\right)=-\cos 2\left(\frac{\pi}{4}\right)=-\cos \frac{\pi}{2}=0 \end{gathered} $$

Hence, the correct answer is option: 'C'.

Relevant Questions