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Question
Use De Moivres theorem and simplify the following:
`(cos (7pi)/13 + "i"sin (7pi)/13)^4/(cos (4pi)/13 - "i"sin (4pi)/13)^6`
Solution
`(cos (7pi)/13 + "i"sin (7pi)/13)^4/(cos (4pi)/13 - "i"sin (4pi)/13)^6`
= `(cos (7pi)/13 + "i"sin (7pi)/13)^4/([cos ((-4pi)/13) + "i"sin ((-4pi)/13)]^6)` ...[∵ sin (– θ) = – sin θ, cos (– θ) = cos θ]
= `(cos pi + "i" sin pi)^(7/13 xx 4)/(cos pi + "i"sin pi)^((-4)/13 xx 6)` ...[∵ cos n θ + i sin n θ = (cos θ + i sin θ)n]
= `(cos pi + "i" sin pi)^(7/13 xx 4) (cos pi + "i"sin pi)^(4/13 xx 6)`
= `(cos pi + "i" sin pi)^(28/13 + 24/13)`
= (cos π + i sin π)4
= cos 4π + i sin 4π ...[∵ (cos θ + i sin θ)n = (cos n θ + i sin n θ)]
= 1 + i(0)
= 1
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