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Let P=tanθ2+2tanθ+4tan2θ+8cot4θ P = \tan \dfrac \theta2 + 2\tan \theta + 4\tan2\theta + 8\cot4\theta P=tan2θ+2tanθ+4tan2θ+8cot4θ and given that 11−cosθ+isinθ \dfrac1{1- \cos\theta + i \sin\theta} 1−cosθ+isinθ1 can be expressed in the form of a+bia + bia+bi, where aaa and bbb are real numbers with i=−1i = \sqrt{-1} i=−1. Compute a⋅bP \dfrac{a\cdot b}{ P } Pa⋅b.
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