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Geometry Level 5

Let \( P = \tan \dfrac \theta2 + 2\tan \theta + 4\tan2\theta + 8\cot4\theta \) and given that \( \dfrac1{1- \cos\theta + i \sin\theta} \) can be expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers with \(i = \sqrt{-1} \). Compute \( \dfrac{a\cdot b}{ P } \).