Question

If $cos x = -\frac{4}{5}$, where $x\in\left[0, \pi\right]$, 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)

$\frac{1}{\sqrt{10}}$

Answer 2

Given, $\cos x=-\frac{4}{5}, x \in[0, \pi]$
$\because \cos x=2 \cos ^{2} \frac{x}{2}-1$
$\therefore 2 \cos ^{2} \frac{x}{2}=1+\cos x=1-\frac{4}{5}=\frac{1}{5}$
$\Rightarrow \cos ^{2} \frac{x}{2}=\frac{1}{10}$
$\Rightarrow \cos \frac{x}{2}=\frac{1}{\sqrt{10}}$
$\begin{bmatrix}\because \cos x \text { is negative, } \\\ \therefore \quad x \in\left(\frac{\pi}{2}, \pi\right] \Rightarrow \frac{x}{2} \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right]\end{bmatrix}$