Question

If $z=i{\log }_e(2-\sqrt{3})$, then find the value of $\cos z$.

• A. $2$
• B. $-2$
• C. $2i$
• D. $-2i$

Answer 1

The correct option is A $2$

Given, $z=i{\log }_e(2-\sqrt{3})$
We know $\cos z=\frac{e^{iz}+e^{-iz}}{2}$
Therefore, $\cos z=\frac{e^{i^2}{\log }_e(2-\sqrt{3})+e^{-i^2}{\log }_e(2-\sqrt{3})}{2}$
$=\frac{e^{{\log }_e}(2-\sqrt{3}{)}^{-1}+e^{{\log }_e}(2-\sqrt{3})}{2}$
$=\frac{(2-\sqrt{3}{)}^{-1}+(2-\sqrt{3})}{2}$
$=\frac{1}{2}[\frac{1}{2-\sqrt{3}}+2-\sqrt{3}]$
$=\frac{1}{2}[2+\sqrt{3}+2-\sqrt{3}]=2$