k = {Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ], Cos[θ]} ;
n = {Cos[α], Cos[β], Cos[γ]} ;
(*k.k // Simplify*)
kp = k - 2 n (k.n) // Simplify ;
thetap = ArcCos[kp.{0,0,1}] // FullSimplify 
phip = ArcTan[ kp.{0,1,0}/ kp.{1,0,0}]