If operatorname{Sinh}^{-1}(2)+operatorname{Si | vedclass

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Question

(D) Given $\operatorname{Sinh}^{-1}(2)+\operatorname{Sinh}^{-1}(3)=\alpha$. Let $x = \operatorname{Sinh}^{-1}(2)$ and $y = \operatorname{Sinh}^{-1}(3)

Vedclass · Accepted Answer

$2 \sqrt{10}+3 \sqrt{5}$

Vedclass · Answer

(D) Given $\operatorname{Sinh}^{-1}(2)+\operatorname{Sinh}^{-1}(3)=\alpha$. Let $x = \operatorname{Sinh}^{-1}(2)$ and $y = \operatorname{Sinh}^{-1}(3)$. Then $\sinh x = 2$ and $\sinh y = 3$. We know that $\cosh^2 x - \sinh^2 x = 1$,so $\cosh x = \sqrt{1 + \sinh^2 x} = \sqrt{1 + 2^2} = \sqrt{5}$. Similarly,$\cosh y = \sqrt{1 + \sinh^2 y} = \sqrt{1 + 3^2} = \sqrt{10}$. We need to find $\sinh \alpha = \sinh(x+y)$. Using the identity $\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y$: $\sinh \alpha = (2)(\sqrt{10}) + (\sqrt{5})(3) = 2 \sqrt{10} + 3 \sqrt{5}$.

If $\operatorname{Sinh}^{-1}(2)+\operatorname{Sinh}^{-1}(3)=\alpha$,then $\sinh \alpha=$

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