Question

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