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Question

Aθ2+Aϕ22(Aθ+ϕ)2 equals

A
2AθAϕ
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B
Aθ+Aϕ
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C
AθAϕ
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D
none of these
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Solution

The correct option is C AθAϕ
Aθ(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(βγ)+sin(αγ)+sin(γβ)
Aθ(α,β,γ)=sin(βγ)+sin(αγ)+sin(γβ)
Aθ is independent of θ
Similarly,
Aϕ and Aθ+ϕ are independent of ϕ and θ+ϕ respectively.
Aθ2+Aϕ22(Aθ+ϕ)2=0
AθAϕ=0
Hence, option C.

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