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

Answer: (A)

Given, $z=i{\log }_e(2-\sqrt{3})\because \cos z=\frac{e^{jz}+e^{-iz}}{2}=\frac{e^{i^2{\log }_e(2-\sqrt{3})}+e^{{\log }_e(2-\sqrt{3})}}{2}$
$=\frac{e^{{\log }_e(2-\sqrt{3}{)}^{-1}+e^{{\log }_e(2-\sqrt{3})}}}{2}=\frac{(2-\sqrt{3}{)}^{-1}+(2-\sqrt{3})}{2}=\frac{1}{2}\left[\frac{1}{2-\sqrt{3}}+2-\sqrt{3}\right]$
$=\frac{1}{2}[2+\sqrt{3}+2-\sqrt{3}]=2$