Question

Prove:${\cos }^{-1}(-x)=\pi -{\cos }^{-1}\left(x\right)$ for all $x\in [-1,1]$

Answer 1

Since $x\in [-1,1]$

$-x\in [-1,1]$

Let ${\cos }^{-1}(-x)=\theta$ so $\theta \in [0,\pi ]$

$\Rightarrow -x=\cos \theta =\cos (\pi -\theta )$

Since $\pi -\theta \in [0,\pi ]$

$\Rightarrow {\cos }^{-1}\cos (\pi -\theta )=\pi -\theta$

$\Rightarrow {\cos }^{-1}\cos x=\pi -\theta$

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

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

Hence proved.