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Question

Prove that:
(i) sin α+sin β+sin γ-sin (α+β+γ)=4 sin α+β2 sin β+γ2 sin γ+α2

(ii) cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos B cos C

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Solution

(i)Consider LHS: sin α + sin β + sin γ - sin (α + β + γ)= 2sinα + β2 cos α - β2 + 2cos γ + α + β + γ2 sin γ - α - β - γ2=2sinα+β2cosα-β2 + 2cos2γ+α+β2sin-α-β2=2sinα+β2cosα-β2 + 2cos2γ+α+β2sin-α+β2=2sinα+β2cosα-β2 - cos2γ+α+β2=2sinα+β2-2sinα-β+2γ+α+β4 sinα-β-2γ-α-β4=2sinα+β2-2sinα+γ2 sin-β-γ2=2sinα+β22sinα+γ2 sinβ+γ2=4sinα+β2 sinα+γ2 sinβ+γ2= RHSHence, LHS=RHS.


(ii)Consider LHS:cos (A+B+C) + cos (A-B+C) + cos (A+B-C) + cos (-A+B+C)=2cos A+B+C+A-B+C2 cos A+B+C-A+B-C2 + 2cos A+B-C-A+B+C2 cos A+B-C+A-B-C2=2cosA+C cos B+2cos B cosA-C=2cos Bcos A+C + cos A-C=2cos B2cos A+C+A-C2 cos A+C-A+C2=2cos B2cos A cos C=4cos A cos B cos C= RHSHence, LHS=RHS.

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