KCET · Maths · Differentiation
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 \(\frac{1}{\sqrt{3}}\)
- C 0
- D \(-\sqrt{3}\)
Answer & Solution
Correct Answer
(C) 0
Step-by-step Solution
Detailed explanation
Given, \(f(x)=\frac{\sin ^{2} x}{1+\cot x}+\frac{\cos ^{2} x}{1+\tan x}\)
\[
\begin{gathered}
f(x)=\frac{\sin ^{2} x}{1+\frac{1}{\tan x}}+\frac{\cos ^{2} x}{1+\tan x} \\
f(x)=\frac{\tan x \cdot \sin ^{2} x+\cos ^{2} x}{(1+\tan x)}=\frac{\sin ^{3} x+\cos ^{3} x}{(\cos x+\sin x)} \\
f(x)=\frac{(\sin x+\cos x)}{(\sin x+\cos x)}\left(\sin ^{2} x+\cos ^{2} x\right. \\
f^{\prime}\left(\frac{\pi}{4}\right)=-\cos 2\left(\frac{\pi}{4}\right)=-\cos \frac{\pi}{2}=0
\end{gathered}
\]
\[
\begin{gathered}
f(x)=\frac{\sin ^{2} x}{1+\frac{1}{\tan x}}+\frac{\cos ^{2} x}{1+\tan x} \\
f(x)=\frac{\tan x \cdot \sin ^{2} x+\cos ^{2} x}{(1+\tan x)}=\frac{\sin ^{3} x+\cos ^{3} x}{(\cos x+\sin x)} \\
f(x)=\frac{(\sin x+\cos x)}{(\sin x+\cos x)}\left(\sin ^{2} x+\cos ^{2} x\right. \\
f^{\prime}\left(\frac{\pi}{4}\right)=-\cos 2\left(\frac{\pi}{4}\right)=-\cos \frac{\pi}{2}=0
\end{gathered}
\]
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