MHT CET · Maths · Differentiation
If \(y=\tan ^{-1}(\sec x-\tan x)\), then \(\frac{d y}{d x}=\)
- A \(2\)
- B \(-\frac{1}{2}\)
- C \(\frac{1}{2}\)
- D \(-2\)
Answer & Solution
Correct Answer
(B) \(-\frac{1}{2}\)
Step-by-step Solution
Detailed explanation
\(y=\tan ^{-1}(\sec x-\tan x)=\tan ^{-1}\left(\frac{1-\sin x}{\cos x}\right)=\) \(\tan ^{-1}\left(\frac{\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}}\right) \)
\( \Rightarrow y=\tan ^{-1}\left(\frac{1-\tan \frac{x}{2}}{1+\tan \frac{x}{2}}\right)=\tan ^{-1}\left(\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)\right)\) \(=\frac{\pi}{4}-\frac{x}{2} \)
\( \Rightarrow \frac{d y}{d x}=\frac{-1}{2}\)
\( \Rightarrow y=\tan ^{-1}\left(\frac{1-\tan \frac{x}{2}}{1+\tan \frac{x}{2}}\right)=\tan ^{-1}\left(\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)\right)\) \(=\frac{\pi}{4}-\frac{x}{2} \)
\( \Rightarrow \frac{d y}{d x}=\frac{-1}{2}\)
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