AP EAMCET · Maths · Differentiation
If \(y=\sin \left(2 \tan ^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right) x=\cos 2 \theta\), then \(\frac{d y}{d x}=\)
- A \(\frac{x}{\sqrt{1-x^2}}\)
- B \(-\cot 2 \theta\)
- C \(\tan 2 \theta\)
- D \(\frac{-x}{2 \sqrt{1-x^2}}\)
Answer & Solution
Correct Answer
(B) \(-\cot 2 \theta\)
Step-by-step Solution
Detailed explanation
\(y=\sin \left(2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right), x=\cos 2 \theta\) Then, \(\frac{1-x}{1+x}=\frac{1-\cos 2 \theta}{1+\cos 2 \theta}=\frac{2 \sin ^2 \theta}{2 \cos ^2 \theta}=\tan ^2 \theta\) Then, \(\tan ^{-1} \sqrt{\frac{1-x}{1+x}}=\tan ^{-1}(\tan \theta)=\theta\) Now,…
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