Let $\theta$ be an angle such that $\tan \theta = 2.$  Simplify
\[\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}.\]
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<|im_start|>user
Let $\theta$ be an angle such that $\tan \theta = 2.$  Simplify
\[\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2i
Let $\theta$ be an angle such that $\tan \theta = 2.$  Simplify
\[\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}.\]
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