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Question

Prove the following:
1+sin Acos A = 1+sin A+cos A1+cos A-sin A

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Solution

RHS =1+sin A+cos A1+cos A -sin A = 1+cos A+sin A1+cos A-sin A= 1+cos A+sin A1+cos A-sin A×1+cos A+sin A1+cos A+sin A= {1+cos A+sin A}21+cos A2-sin2A =1+cos A2+sin2A+2sin A1+cos A1+cos2A+2cos A-sin2A=1+cos2A+2cos A+sin2A+2(1+cosA)sinA1+cos2A+2cosA-sin2A=1+cos2A+sin2A+2cos A+2(1+cosA)sinA1-sin2A+cos2A+2cosA= 1+1+2cosA+2(1+cosA)sin Acos2A+cos2A+2cos A=2+2cos A+2(1+cos A) sin A2cos2A+2cos A

=2(1+cosA) +2(1+cosA)sinA2cosA(1+cosA)=2(1+cosA)(1+sinA)2cosA(1+cosA)=1+sinAcosA=LHSHence, proved.

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