sinh^{-1}(-2) + operatorname{cosech}^{-1}(-2) | vedclass

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Question

(A) We use the logarithmic definitions of inverse hyperbolic functions: $\sinh^{-1}(x) = \log(x + \sqrt{x^2+1})$ $\operatorname{cosech}^{-1}(x) = \log

Vedclass · Accepted Answer

$\log \left(\frac{7-3 \sqrt{5}}{2 \sqrt{3}}\right)$

Vedclass · Answer

(A) We use the logarithmic definitions of inverse hyperbolic functions: $\sinh^{-1}(x) = \log(x + \sqrt{x^2+1})$ $\operatorname{cosech}^{-1}(x) = \log\left(\frac{1}{x} + \sqrt{\frac{1}{x^2}+1}\right)$ $\coth^{-1}(x) = \frac{1}{2} \log\left(\frac{x+1}{x-1}\right)$ Substituting $x = -2$: $\sinh^{-1}(-2) = \log(-2 + \sqrt{5})$ $\operatorname{cosech}^{-1}(-2) = \log\left(-\frac{1}{2} + \sqrt{\frac{1}{4}+1}\right) = \log\left(\frac{-1+\sqrt{5}}{2}\right)$ $\coth^{-1}(-2) = \frac{1}{2} \log\left(\frac{-2+1}{-2-1}\right) = \frac{1}{2} \log\left(\frac{-1}{-3}\right) = \frac{1}{2} \log\left(\frac{1}{3}\right) = \log\left(\frac{1}{\sqrt{3}}\right)$ Summing these: $\log(-2 + \sqrt{5}) + \log\left(\frac{\sqrt{5}-1}{2}\right) + \log\left(\frac{1}{\sqrt{3}}\right) = \log\left[(-2 + \sqrt{5}) \cdot \left(\frac{\sqrt{5}-1}{2}\right) \cdot \frac{1}{\sqrt{3}}\right]$ $= \log\left[\frac{-2\sqrt{5} + 2 + 5 - \sqrt{5}}{2\sqrt{3}}\right] = \log\left(\frac{7-3\sqrt{5}}{2\sqrt{3}}\right)$

$\sinh^{-1}(-2) + \operatorname{cosech}^{-1}(-2) + \coth^{-1}(-2) = $

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