Find the values of other five trigonometric functions if $\cos x=-\frac{1}{2}, x$ lies in third quadrant.
$\cos x=-\frac{1}{2}$
$\therefore \sec x=\frac{1}{\cos x}=\frac{1}{\left(-\frac{1}{2}\right)}=-2$
$\sin ^{2} x+\cos ^{2} x=1$
$\Rightarrow \sin ^{2} x=1-\cos ^{2} x$
$\Rightarrow \sin ^{2} x=1-\left(-\frac{1}{2}\right)^{2}$
$\Rightarrow \sin ^{2} x=1-\frac{1}{4}=\frac{3}{4}$
$\Rightarrow \sin x=\pm \frac{\sqrt{3}}{2}$
since $x$ lies in the $3^{\text {rd }}$ quadrant, the value of $\sin x$ will be negative.
$\therefore \sin x=-\frac{\sqrt{3}}{2}$
$\cos ec\, x=\frac{1}{\sin x}=\frac{1}{\left(-\frac{\sqrt{3}}{2}\right)}=-\frac{2}{\sqrt{3}}$
$\tan x=\frac{\sin x}{\cos x}=\frac{\left(-\frac{\sqrt{3}}{2}\right)}{\left(-\frac{1}{2}\right)}=\sqrt{3}$
$\cot x=\frac{1}{\tan x}=\frac{1}{\sqrt{3}}$
Find the value of the trigonometric function $\cot \left(-\frac{15 \pi}{4}\right)$
$\frac{{1 + \sin A - \cos A}}{{1 + \sin A + \cos A}} =$
$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + ..... + \cos 180^\circ = $
If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5},$ then
$(A)$ $\tan ^2 x=\frac{2}{3}$ $(B)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$
$(C)$ $\tan ^2 x=\frac{1}{3}$ $(D)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$
If $\tan \theta = \frac{a}{b},$ then $\frac{{\sin \theta }}{{{{\cos }^8}\theta }} + \frac{{\cos \theta }}{{{{\sin }^8}\theta }} = $