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cos A + cos (A+B) + cos (A+2B)+…+ cos [A+(n-1)B] = cos [A+(n-1)B/2] sin nB/2 /sin B/2
Dear Sonam
let
S =cos A + cos (A+B) + cos (A+2B)+…+ cos [A+(n-1)B]
or S = Real part of [eiA + ei(A+B) + ei(A+2B)+…+ ei(A+(n-1)B)]
S = Re [eiA{1+ei(B) + ei(2B)+…+ ei((n-1)B)}]
= Re [eiA{1+ei(B) + {ei(B)}2+{ei(B)}3…+ {ei(B)}n-1) }]
=Re [eiA{{ei(B)}n-1}/{ei(B)-1}]
=Re [eiA{cosnB + isin nB-1}/{cosB + isinB -1}]
= Re [eiA{1-2sin2nB/2 + i2sin nB/2 cosnB/2 -1}/{1-2sin2B/2 + i2sinB/2 cosB/2 -1}]
=Re [eiA{-2sin2nB/2 + i2sin nB/2 cosnB/2 }/{-2sin2B/2 + i2sinB/2 cosB/2 }]
=Re [eiA2isin nB/2{ cosnB/2 +isin nB/2 }/2isinB/2{cosB/2 +isin B/2 }]
=Re [sin nB/2 (cosA+isinA){ cosnB/2 +isin nB/2 }/sinB/2{cosB/2 +isin B/2 }]
=sin (nB/2) /sin(B/2) Re [ cos(A+(n-1)B/2) +isin(A+(n-1)B/2)
compair real part
S =cos [A+(n-1)B/2] sin nB/2 /sin B/2
Please feel free to post as many doubts on our discussion forum as you can.If you find any question Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We are all IITians and here to help you in your IIT JEE preparation.All the best. Regards,Askiitians ExpertsBadiuddin
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