Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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