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

2

The correct answer is Option (3) → 2 ##

$\cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi$

$\Rightarrow \cos^{-1} \alpha = \pi, \cos^{-1} \beta = \pi \ \& \ \cos^{-1} \gamma = \pi$

$∴\alpha = \beta = \gamma = -1$

$\alpha(\beta + \gamma) - \beta(\gamma + \alpha) + \gamma(\alpha + \beta)$

$= (-1)(-1 - 1) - (-1)(-1 - 1) + (-1)(-1 - 1)$

$= 2 - 2 + 2 = 2$