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Geometry Level 3

Given that tanθ+cotθ=2\tan \theta +\cot \theta =2. And

2A=n=122016tan2nθ+cot2nθ\Large 2^A = \displaystyle \sum^{2^{2016}}_{n=1}\color{#20A900}{\tan}^{\color{#D61F06}{ 2}^{\color{#3D99F6}{n}}}\theta+\color{#20A900}{\cot}^{\color{#D61F06}{2}^{ \color{#3D99F6}{n}}}\theta

Compute AA.

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