Find the values of other five trigonometric functions if $cos x=- rac{1}{2}$ ,

$\cos x=-\frac{1}{2}$

$\therefore \sec x=\frac{1}{\cos x}=\frac{1}{\left(-\frac{1}{2}\right)}=-2$

$\sin ^{2} x+\cos ^{2} x=1$

$\Rightarrow \sin ^{2} x=1-\cos ^{2} x$

$\Rightarrow \sin ^{2} x=1-\left(-\frac{1}{2}\right)^{2}$

$\Rightarrow \sin ^{2} x=1-\frac{1}{4}=\frac{3}{4}$

$\Rightarrow \sin x=\pm \frac{\sqrt{3}}{2}$

Since $x$ lies in the $3^{\text {rd }}$ quadrant, the value of $\sin x$ will be negative.

$\therefore \sin x=-\frac{\sqrt{3}}{2}$

$\operatorname{cosec} x=\frac{1}{\sin x}=\frac{1}{\left(-\frac{\sqrt{3}}{2}\right)}=-\frac{2}{\sqrt{3}}$

$\tan x=\frac{\sin x}{\cos x}=\frac{\left(-\frac{\sqrt{3}}{2}\right)}{\left(-\frac{1}{2}\right)}=\sqrt{3}$

$\cot x=\frac{1}{\tan x}=\frac{1}{\sqrt{3}}$