(({{Cos[ϕ] , -Sin[ϕ], 0}, {Sin[ϕ], Cos[ϕ] , 0}, {0, 0, 1}})).(({{Cos[θ] , 0, Sin[θ]}, {0, 1, 0}, {-Sin[θ], 0, Cos[θ]}})).(({{Cos[ψ] , -Sin[ψ], 0}, {Sin[ψ], Cos[ψ] , 0}, {0, 0, 1}}))

Det[(({{Cos[ϕ] , -Sin[ϕ], 0}, {Sin[ϕ], Cos[ϕ] , 0}, {0, 0, 1}})).(({{Cos[θ] , 0, Sin[θ]}, {0, 1, 0}, {-Sin[θ], 0, Cos[θ]}})).(({{Cos[ψ] , -Sin[ψ], 0}, {Sin[ψ], Cos[ψ] , 0}, {0, 0, 1}}))]

Simplify[
      Cos[θ]^2 Cos[ϕ]^2 Cos[ψ]^2 + 
        Cos[ϕ]^2 Cos[ψ]^2 Sin[θ]^2 + 
        Cos[θ]^2 Cos[ψ]^2 Sin[ϕ]^2 + 
        Cos[ψ]^2 Sin[θ]^2 Sin[ϕ]^2 + 
        Cos[θ]^2 Cos[ϕ]^2 Sin[ψ]^2 + 
        Cos[ϕ]^2 Sin[θ]^2 Sin[ψ]^2 + 
        Cos[θ]^2 Sin[ϕ]^2 Sin[ψ]^2 + 
        Sin[θ]^2 Sin[ϕ]^2 Sin[ψ]^2]

Simplify[
      Dot[{Cos[θ] Cos[ϕ] Cos[ψ] - 
            Sin[ϕ] Sin[ψ], -Cos[ψ] Sin[ϕ] - 
            Cos[θ] Cos[ϕ] Sin[ψ], 
          Cos[ϕ] Sin[θ]}, {Cos[θ] Cos[ϕ] Cos[ψ] - Sin[ϕ] Sin[ψ], -Cos[ψ] Sin[ϕ] - 
            Cos[θ] Cos[ϕ] Sin[ψ], 
          Cos[ϕ] Sin[θ]}]]

Simplify[
      Dot[{Cos[θ] Cos[ϕ] Cos[ψ] - 
            Sin[ϕ] Sin[ψ], -Cos[ψ] Sin[ϕ] - 
            Cos[θ] Cos[ϕ] Sin[ψ], 
          Cos[ϕ] Sin[θ]}, {Cos[θ] Cos[ψ] Sin[ϕ] + Cos[ϕ] Sin[ψ], 
          Cos[ϕ] Cos[ψ] - Cos[θ] Sin[ϕ] Sin[ψ], 
          Sin[θ] Sin[ϕ]}]]