Prove that ${\cos }^{-1}\left(-x\right)=\pi -{\cos }^{-1}\left(x\right)$, $-1\le x\le 1$. 

Let $\cos A=x$ then $\cos (\pi -A)=-\cos A=-x$

Hence, ${\cos }^{-1}x=A$ 

${\cos }^{-1}(-x)=\pi -A$ 

Adding them,

$$\begin{gathered} {\cos }^{-1}x+{\cos }^{-1}(-x)=A+\pi -A\Rightarrow {\cos }^{-1}x+{\cos }^{-1}(-x)=\pi \\ \Rightarrow {\cos }^{-1}(-x)=\pi -{\cos }^{-1}(x) \end{gathered}$$