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Geometry Level 5

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

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