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Question

Prove that : 1+cosA+sinA1+cosAsinA=1+sinAcosA

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Solution

We have, 1+cosA+sinA1+csosAsinA

=1+cosA+sinA1+cosAsinA×(1+cosA)+sinA(1+cosA)+sinA

=((1+cosA)+sinA)2(1+cosA)2sin2A

=(1+cosA)2+sin2A+2(1+cosA)sinA1+cos2A+2cosA1+cos2A

=1+cos2A+sin2A+2cosA+2sinA+2sinAcosA2cos2A+2cosA

=1+cos2A+sin2A+2cosA+2sinA+2sinAcosA2cos2A+2cosA

1+1+2cosA+2sinA+2sinAcosA2cosA(1+cosA)

2+2cosA+2sinA+2sinAcosA2cosA(1+cosA)

1+cosA+sinA+sinAcosAcosA(1+cosA)

=1+sinA+cosA(1+sinA)cosA(1+cosA)

=(1+sinA)(1+cosA)cosA(1+cosA)

=1+sinAcosA

Hence proved.

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