Prove that- 1-1-2cos- -1-3-2-4cos 2- -1-3-5-2-4-6cos 3- - 2sin-2 -1-2 - cos- -4-isin- -4 -

The given series is C and choose sine series S= 1/2sinα +1.3/2.4sin 2α +1.3.5/2.4.6sin 3α +⋯ ∴ C+iS= 1+12e^iα+12.321.2e^2iα+12.32.521.2.3e^3iα+⋯ = ...

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Byju's Answer

Standard XII

Mathematics

Complex Numbers

prove that: ...

Question

prove that:
$1 + \frac{1}{2} cos α + \frac{1.3}{2.4} cos 2 α + \frac{1.3.5}{2.4.6} cos 3 α + \dots = {(2 sin \frac{α}{2})}^{- 1 / 2} [cos \frac{π - α}{4} + i sin \frac{π - α}{4}]$ .

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Solution

The given series is C and choose sine series
$S = \frac{1}{2} sin α + \frac{1.3}{2.4} sin 2 α + \frac{1.3.5}{2.4.6} sin 3 α + \dots$
$∴ C + i S = 1 + \frac{1}{2} e^{i α} + \frac{\frac{1}{2} . \frac{3}{2}}{1.2} e^{2 i α} + \frac{\frac{1}{2} . \frac{3}{2} . \frac{5}{2}}{1.2.3} e^{3 i α} + \dots$
$= 1 + \frac{1}{2} e^{i α} + \frac{\frac{1}{2} (\frac{1}{2} + 1)}{2!} e^{2 i α} + \frac{\frac{1}{2} (\frac{1}{2} + 1) (\frac{1}{2} + 2)}{3!} e^{3 i α} + \dots$
$= {(1 - e^{i a})}^{- 1 / 2} = {(1 - cos α - i sin α)}^{- 1 / 2}$
$= {(2 {sin}^{2} \frac{α}{2} - 2 i sin \frac{α}{2} cos \frac{α}{2})}^{- 1 / 2}$
$= {(2 sin \frac{α}{2})}^{- 1 / 2} {[sin \frac{α}{2} - i cos \frac{α}{2}]}^{- 1 / 2}$
$= {(2 sin \frac{α}{2})}^{- 1 / 2} {[cos (\frac{π}{2} - \frac{α}{2}) - i sin (\frac{π}{2} - \frac{α}{2})]}^{- 1 / 2}$
$= {(2 sin \frac{α}{2})}^{- 1 / 2} [cos \frac{π - α}{4} + i sin \frac{π - α}{4}]$

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Similar questions

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prove that:1+12cosα+1.32.4cos2α+1.3.52.4.6cos3α+⋯=(2sinα2)−1/2[cosπ−α4+isinπ−α4].

prove that:
$1 + \frac{1}{2} cos α + \frac{1.3}{2.4} cos 2 α + \frac{1.3.5}{2.4.6} cos 3 α + \dots = {(2 sin \frac{α}{2})}^{- 1 / 2} [cos \frac{π - α}{4} + i sin \frac{π - α}{4}]$ .