Question

If $cos\,x=\frac{-4}{5}$, where $x\in\left[0, \pi/2\right]$, then the value of $cos\left(\frac{x}{2}\right)$ is equal to

• A. $\frac{1}{10}$
• B. $\frac{2}{5}$
• C. $\frac{1}{\sqrt{10}}$
• D. $-\frac{2}{5}$

Answer 1

Answer: (C)

We have, $cos\,x=\frac{-4}{5}$, $x \in\left[0, \frac{\pi}{2}\right]$
$cos\,x=2\,cos^{2} \frac{x}{2}-1$
$\Rightarrow 2\,cos^{2} \frac{x}{2}=\frac{1}{5}$
$\Rightarrow cos^{2} \frac{x}{2}=\frac{1}{10}$
$\Rightarrow cos \frac{x}{2}=\pm \frac{1}{\sqrt{10}}$
$cos \frac{x}{2}$ is positive in $I$ quadrant $\left(0 < \frac{x}{2} < \frac{\pi}{4}\right)$
$\therefore cos \frac{x}{2}=\frac{1}{\sqrt{10}}$