Question

$\sinh \left(\log \right(2+\sqrt{5}\left)\right)=$
(Note : $\sinh \left(x\right)=\frac{e^x-e^{-x}}{2}$ , logarithm is to the natural base e )

• A. $2$
• B. $3$
• C. $2-\sqrt{5}$
• D. $2+\sqrt{5}$

Answer 1

The correct option is A $2$

$\sinh x=\frac{e^x-e^{-x}}{2}$

$\sinh \left(log\right(2+\sqrt{5}\left)\right)=\frac{e^{log(2+\sqrt{5})}-e^{-log(2+\sqrt{5})}}{2}$

$=\frac{2+\sqrt{5}-\frac{1}{2+\sqrt{5}}}{2}$

$=\frac{2+\sqrt{5}-\left(\frac{\sqrt{5}-2}{5-4}\right)}{2}$

$=\frac{2+\sqrt{5}-(\sqrt{5}-2)}{2}$

$=\frac{4}{2}$

$=2$.