The correct option is
C Aθ−AϕAθ(α,β,γ)=∣∣
∣
∣∣cos(α+θ)sin(α+θ)1cos(β+θ)sin(β+θ)1cos(γ+θ)sin(γ+θ)1∣∣
∣
∣∣=cos(α+θ)(sin(β+θ)−sin(γ+θ))−sin(α+θ)(cos(β+θ)−cos(γ+θ))+(cos(β+θ)sin(γ+θ)−sin(β+θ)cos(γ+θ))
=sin(β−γ)+sin(α−γ)+sin(γ−β)
∴Aθ(α,β,γ)=sin(β−γ)+sin(α−γ)+sin(γ−β)
Aθ is independent of θ
Similarly,
Aϕ and Aθ+ϕ are independent of ϕ and θ+ϕ respectively.
∴Aθ2+Aϕ2−2(Aθ+ϕ)2=0
Aθ−Aϕ=0
Hence, option C.