Find all other trigonometrical ratios if \(\sin x=\frac{-2\sqrt6}{5}\) and x lies in III quadrant?

Prove that \(\sin^2\frac{\pi}{6}+\cos^2\frac{\pi}{3}-\tan^2\frac{\pi}{4}=-\frac12\)

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

Write \(\tan ^{ -1 }{ \left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) } ,\left| x \right| >1\) in the simplest form.

Prove that \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}\)