AP EAMCET · Maths · Trigonometric Ratios & Identities
If \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\), then \(\log \left(\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right)=\)
- A \(\tanh ^{-1}\left(\tan \frac{\theta}{2}\right)\)
- B \(2 \tanh ^{-1}\left(\tan \frac{\theta}{2}\right)\)
- C \(\operatorname{coth}^{-1}\left(\tan \frac{\theta}{2}\right)\)
- D \(2 \operatorname{coth}^{-1}\left(\tan \frac{\theta}{2}\right)\)
Answer & Solution
Correct Answer
(B) \(2 \tanh ^{-1}\left(\tan \frac{\theta}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{array}{rlrl} \text {Let } & \log \left(\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right) =x \\ \Rightarrow & \tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right) =e^x \\ \Rightarrow & \frac{1+\tan \frac{\theta}{2}}{1-\tan \frac{\theta}{2}} =e^x \end{array}\) On…
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