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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 $\dfrac{a\cdot b}{ P }$ .