If Δ=sin⁡θcos⁡ϕsin⁡θsin⁡ϕcos⁡θcos⁡θcos⁡ϕcos⁡θsin⁡ϕ−sin⁡θ−sin⁡θsin⁡ϕsin⁡θcos⁡ϕ0 then

Applying C1→C1−(cot⁡ϕ)C2, we getΔ=0sin⁡θsin⁡ϕcos⁡θ0cos⁡θsin⁡ϕ−sin⁡θ−sin⁡θ/sin⁡ϕsin⁡θcos⁡ϕ0=−sin⁡θsin⁡ϕ−sin⁡ϕsin2⁡θ−cos2⁡θsin⁡ϕ               [expanding alone C1]=sin⁡θ, which is independent of ϕ.Also, dΔdθ=cos⁡θ⇒dΔdθθ=π/2=cos⁡(π/2)=0