Question

If ${\cos }^{-1}\alpha +{\cos }^{-1}\beta +{\cos }^{-1}\gamma =3\pi$ then find $\alpha (\beta +\gamma )+\beta (\gamma +\alpha )+\gamma (\alpha +\beta )$

Answer 1

The maximum value of ${\cos }^{-1}x$ is $\pi$
Therefore for the given sum to be $3\pi$, all the individual terms must be $\pi$.
$⟹\alpha =\beta =\gamma =-1$
$\therefore \alpha (\beta +\gamma )+\beta (\gamma +\alpha )+\gamma (\alpha +\beta )=-1(-2)\times 3=6$