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Question
Use De Moivres theorem and simplify the following:
`(cos5theta + "i"sin5theta)/((cos3theta - "i"sin3theta)^2`
Solution
`(cos5theta + "i"sin5theta)/((cos3theta - "i"sin3theta)^2`
= `(cos 5theta + "i" sin 5theta)/[cos(-3theta) + "i" sin(-3theta)]^2` ...[∵ cos(– θ) = cos θ, sin(– θ) = – sin θ]
= `(cos theta + "i" sin theta)^5/((cos theta + "i" sin theta)^(-3 xx 2)` ...[∵ (cos θ + i sin θ)n = cos nθ + i sin nθ]
= (cos θ + i sin θ)5+6
= (cos θ + i sin θ)11
= cos 11θ + i sin 11θ
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