Question

${\sinh }^{−1}2+{\sinh }^{−1}3=x⇒\cosh x$ is equal to

• A. $\frac{1}{2}(3\sqrt{5}+2\sqrt{10})$
• B. $\frac{1}{2}(3\sqrt{5}−2\sqrt{10})$
• C. $\frac{1}{2}(12+2\sqrt{50})$
• D. $\frac{1}{2}(12−2\sqrt{50})$

Answer 1

Answer: (C)

Given, ${\sinh }^{−1}2+{\sinh }^{−1}3=x$
$⇒\cosh \left({\sinh }^{−1}2+{\sinh }^{−1}3\right)=\cosh x$
$⇒\cosh \left({\sinh }^{−1}2\right)\cosh \left({\sinh }^{−1}3\right)$
$+\sinh ({\sinh }^{−1}$ 2) $\sinh \left({\sinh }^{−1}3\right)$
$=\cosh x$
$⇒\cosh x=\cosh \left({\cosh }^{−1}\sqrt{1+2^2}\right)×$
$\cosh \left({\cosh }^{−1}\sqrt{1+3^2}\right)+2×3$
$⇒\cosh x=(\sqrt{5}\sqrt{10}+6)×\frac{2}{2}$
$=\frac{1}{2}(12+2\sqrt{50})$