AP EAMCET · Maths · Trigonometric Ratios & Identities
If \(x=-\frac{1}{2}, \sinh ^{-1} x+\operatorname{cosech}^{-1} x=\)
- A \(\log _e\left(\frac{7-3 \sqrt{5}}{2}\right)\)
- B \(\log _6\left(\frac{3+\sqrt{5}}{2}\right)\)
- C \(\log _e\left[\frac{(\sqrt{5}-1)(2+\sqrt{3})}{2}\right]\)
- D \(\log _e\left[\frac{(\sqrt{5}+1)(2+\sqrt{3})}{2}\right]\)
Answer & Solution
Correct Answer
(A) \(\log _e\left(\frac{7-3 \sqrt{5}}{2}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Given, } x=-\frac{1}{2} \\ & \because \\ & \text { and } \quad \sinh ^{-1} x=\ln \left(x+\sqrt{x^2+1}\right) \\ & \therefore \sinh ^{-1} x+\operatorname{cosech}^{-1} x=\ln \left(\frac{1}{x}+\sqrt{\frac{1}{x^2}+1}\right) \\ & =\ln…
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