Question

If tan $x=\frac{3}{4},$ $\pi <x<\frac{3\pi }{2}$ then the value of $cos\frac{x}{2}$ is

• A. $\frac{3}{\sqrt{10}}$
• B. $-\frac{3}{\sqrt{10}}$
• C. $-\frac{1}{\sqrt{10}}$
• D. $\frac{1}{\sqrt{10}}$

Answer 1

Answer: (C)

Given, $\tan x=\frac{3}{4},\pi <x<\frac{3\pi }{2}$
$\Rightarrow \cos x=-\frac{4}{5}$
$\because 1+\cos x=2{\cos }^2\left(\frac{x}{2}\right)$
$\therefore 1-\frac{4}{5}=2{\cos }^2\left(\frac{x}{2}\right)$
$\Rightarrow \frac{1}{5}=2{\cos }^2\left(\frac{x}{2}\right)$
$\Rightarrow {\cos }^2\left(\frac{x}{2}\right)=\frac{1}{10}$
$\Rightarrow \cos \left(\frac{x}{2}\right)=\pm \frac{1}{\sqrt{10}}$
$\cos \left(\frac{x}{2}\right)=-\frac{1}{\sqrt{10}}$
$\left[\because \frac{\pi }{2}<\frac{x}{2}<\frac{3\pi }{4}\right]$