Q103 / 170 · Manipal (MET) 2020 · Mathematics
If $Z=i \log (2-\sqrt{3})$, then the value of $\cos Z$ will be
- A$i$
- B$2 i$
- C$1$
- Correct answerD$2$
Solution
Given, $Z=i \log (2-\sqrt{3})$
$\Rightarrow, \cos Z_4=\cos [i \log (2-\sqrt{3})]$
$=\cosh \{\log (2-\sqrt{3})\}$
$[\because \cos i x=\cosh x$, for $x \in R]$
$\begin{aligned} & =\frac{e^{\log (2-\sqrt{3})}+e^{-\log (2-\sqrt{3})}}{2} \\ & \quad\left[\because \cosh x=\frac{e^x+e^{-x}}{2}\right]\end{aligned}$
$=\frac{e^{\log (2-\sqrt{3})}+e^{\log \left(\frac{1}{2-\sqrt{3}}\right)}}{2}$
$\begin{aligned} & =\frac{2-\sqrt{3}+\frac{1}{2-\sqrt{3}}}{2} \\ & =\frac{(2-\sqrt{3})^2+1}{2(2-\sqrt{3})}=\frac{4+3-4 \sqrt{3}+1}{2(2-\sqrt{3})} \\ & =\frac{8-4 \sqrt{3}}{2(2-\sqrt{3})}=\frac{4(2-\sqrt{3})}{2(2-\sqrt{3})}=2\end{aligned}$
Hence, the correct answer is option: 'D'.