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Question

Find the real values of θ for which the complex number 1+i cosθ1-2i cosθ is purely real.

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Solution

1+icosθ1-2icosθ=1+icosθ1-2icosθ×1+2icosθ1+2icosθ=1+2icosθ+icosθ-2cosθ1+4cos2θ=1-2cosθ+i3cosθ1+4cos2θFor it to be purely real, the imaginary part must be zero.3cosθ=0This is true for odd multiples of π2.θ=2n+1π2, nZ

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