Given equation: 

${\cos }^{-1}\lambda +{\cos }^{-1}\mu +{\cos }^{-1}\gamma =3\pi$

As. range of ${\cos }^{-1}x$ is $\left[0,\pi \right]$

$\Rightarrow {\cos }^{-1}\lambda =\pi ; {\cos }^{-1}\mu =\pi ; {\cos }^{-1}\gamma =\pi$

Only possible solution of given equation 

$\Rightarrow \lambda =\cos \pi =-1$

$\Rightarrow \mu =\cos \pi =-1$

$\Rightarrow \gamma =\cos \pi =-1$

Value of equation

$\lambda \mu +\mu \gamma +\gamma \lambda =\left(-1\times -1\right)+\left(-1\times -1\right)+\left(-1\times -1\right)$

$= 1+1+1=3$

$\Rightarrow \lambda \mu +\mu \gamma +\gamma \lambda =3$

Hence solved