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

If \(\alpha\) and \(\beta\) are acute angles such that \(\tan\alpha=\frac{m}{m+1}\) and \(\tan\beta=\frac{1}{2m+1}\), prove that \(\alpha+\beta=\frac{\pi}{4}\)

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Prove that \(\frac{\sin5x+\sin3x}{\cos5x+\cos3x}=\tan4x\)

Prove that \(\frac{\sin5x-2\sin3x+sinx}{\cos5x-\cos x}=\tan x\)

Show that \(\cos^{-1}\left(\frac{3}{5}\cos x+\frac45\sin x\right)=x-\tan^{-1}\left(\frac43\right)\)