Question

Find the value of $\cos \left(\frac{x}{2}\right)$, if $\tan x=\frac{5}{12}$ and $x$ lies in third quadrant.

• A. $\frac{5}{\sqrt{13}}$
• B. $\frac{5}{\sqrt{26}}$
• C. $\frac{5}{13}$
• D. $\sqrt{\frac{1}{26}}$

Answer 1

Answer: (D)

Given, $\tan x=\frac{5}{12}$ and $x$ lies in III quadrant.
$\therefore \sin x=\frac{-5}{13} \text { and } \cos x=\frac{-12}{13}$

Now, $\cos x=2 \cos ^2 \frac{x}{2}-1$
$ \Rightarrow \cos ^2 \frac{x}{2}=\frac{1}{2}(\cos x+1) $
$ =\frac{1}{2}\left(\frac{-12}{13}+1\right)=\frac{1}{2}\left(\frac{1}{13}\right)=\frac{1}{26} $
$ \therefore \cos \frac{x}{2}=\sqrt{\frac{1}{26}}$