Q.Prove that $\cos^{-1}(-x)=\pi-\cos^{-1}x$ for $x\in[-1,1]$, and use it to find $\cos^{-1}\!\left(-\dfrac12\right)$.

Two families of elementary identities follow directly from the definition of each principal value branch. The reciprocal identities relate a function to its co-reciprocal partner, e.g. $\sin^{-1}(1/x)=\text{cosec}^{-1}x$ for $|x|\ge1$. The negative-argument identities describe how the inverse function responds to replacing $x$ by $-x$: $\sin^{-1}$, $\tan^{-1}$ and $\text{cosec}^{-1}$ are odd ($\sin^{-1}(-x)=-\sin^{-1}x$), while $\cos^{-1}$, $\sec^{-1}$ and $\cot^{-1}$ satisfy a $\pi$-shift rule instead ($\cos^{-1}(-x)=\pi-\cos^{-1}x$), because their principal branches $[0,\pi]$ and $(0,\pi)$ are not symmetric about the origin the way $\left[-\pi/2,\pi/2\right]$ is.