AP EAMCET · Maths · Inverse Trigonometric Functions
\(\operatorname{Sech}^{-1}(\sin \alpha)=\)
- A \(\log \left(\sin \alpha+\sqrt{\sin ^2 \alpha-1}\right)\)
- B \(\log (\tan \alpha+1)\)
- C \(\log \left(\cot \frac{\alpha}{2}\right)\)
- D \(\log \left(\frac{1+\tan \alpha}{2 \sin \alpha}\right)\)
Answer & Solution
Correct Answer
(C) \(\log \left(\cot \frac{\alpha}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(\operatorname{Sech}^{-1}(\sin \alpha) = \log \left(\frac{1+\sqrt{1-\sin^2 \alpha}}{\sin \alpha}\right)\) \(= \log \left(\frac{1+\cos \alpha}{\sin \alpha}\right)\) \(= \log \left(\frac{2\cos^2 \frac{\alpha}{2}}{2\sin \frac{\alpha}{2}\cos \frac{\alpha}{2}}\right)\)…
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