Prove the ${\sin }^{-1}\left(-\frac{1}{2}\right)+{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\frac{2\mathrm{\pi }}{3}$.

Using principal values, find the value of :

${\cos }^{-1}\left(\frac{1}{2}\right)-2 {\sin }^{-1}\left(-\frac{1}{2}\right)$. (Give answer in how many terms of $\mathrm{\pi })$

Using principal values, find the value of :

${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}\left(-\sqrt{3}\right)$ in how many terms of $\mathrm{\pi }$

If ${\tan }^{-1}\left(1\right)+{\cos }^{-1}\left(-\frac{1}{2}\right)=\frac{\mathrm{k\pi }}{12}$. Then, find the value of $k$.

Using principal values, find the value of :

${\mathrm{cosec}}^{-1}\left(-1\right)+{\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$. (Give answer in how many terms of $\mathrm{\pi })$

Find the value of :

${\tan }^{-1}\left(\tan \frac{5\mathrm{\pi }}{6}\right)+{\cos }^{-1}\left(\cos \frac{13\mathrm{\pi }}{6}\right)$.