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Question

prove that:
1+12cosα+1.32.4cos2α+1.3.52.4.6cos3α+=(2sinα2)1/2[cosπα4+isinπα4].

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Solution

The given series is C and choose sine series
S=12sinα+1.32.4sin2α+1.3.52.4.6sin3α+
C+iS=1+12eiα+12.321.2e2iα+12.32.521.2.3e3iα+
=1+12eiα+12(12+1)2!e2iα+12(12+1)(12+2)3!e3iα+
=(1eia)1/2=(1cosαisinα)1/2
=(2sin2α22isinα2cosα2)1/2
=(2sinα2)1/2[sinα2icosα2]1/2
=(2sinα2)1/2[cos(π2α2)isin(π2α2)]1/2
=(2sinα2)1/2[cosπα4+isinπα4]

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