Question

If $\sin h^{-1}\left(2\right)+\sin h^{-1}\left(3\right)=x$, then $\cos hx$ 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

The correct option is C

$\frac{1}{2}[12+2\sqrt{50}]$

Step1. Applying given condition :

Given $\sin h^{-1}\left(2\right)+\sin h^{-1}\left(3\right)=x$

We now that $\sin h^{-1}\left(x\right)=\ln [x+\sqrt{x^2+1}]$

Then $\sin h^{-1}\left(2\right)+\sin h^{-1}\left(3\right)=\ln [2+\sqrt{4+1}]+\ln [3+\sqrt{9+1}]$

$$\begin{gathered} =\ln (2+\sqrt{5})+\ln (3+\sqrt{10}) \\ =\ln \left[\right(2+\sqrt{5}\left)\right(3+\sqrt{10}\left)\right] \\ =x \end{gathered}$$

Step2. Find the value of $\cos hx$:

Now, $\cos hx=\frac{(e^x+e^{-x})}{2}$

$$\begin{gathered} =\frac{e^{ln\left[\right(2+\sqrt{5}\left)\right(3+\sqrt{10}\left)\right]}+e^{-ln\left[\right(2+\sqrt{5}\left)\right(3+\sqrt{10}\left)\right]}}{2} \\ =\frac{1}{2}\left[(2+\sqrt{5})(3+\sqrt{10})–\frac{1}{(2+\sqrt{5})(3+\sqrt{10})}\right] \end{gathered}$$

On further simplification we get,

$\frac{1}{2}[12+2\sqrt{50}]$

Hence, the correct option is (C).