Question

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

Answer 1

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

Hence, ${\cos }^{-1}x=A$

${\cos }^{-1}(-x)=\pi -A$

Adding them,

${\cos }^{-1}x+{\cos }^{-1}(-x)=A+\pi -A\Rightarrow {\cos }^{-1}x+{\cos }^{-1}(-x)=\pi \Rightarrow {\cos }^{-1}(-x)=\pi -{\cos }^{-1}\left(x\right)$