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P(two|one)=\frac{P(one\cap \textbb{the other})}{P(one)}
\frac{1}{2} \tan\frac{\theta}{2} + \frac{1}{2^2} \tan\frac{\theta}{2^2} + ... + \frac{1}{2^{n+1}} \tan\frac{\theta}{2^{n+1}}}
= \frac{1}{2^n}}\cot\frac{\theta}{2^n} - \cot\theta + \frac{1}{2^{n+1}}} \tan\frac{\theta}{2^{n+1}}
= \frac{1}{2^n}(\frac{1-tan^2\frac{\theta}{2^{n+1}}}{2tan\frac{\theta}{2^{n+1}}} + \frac{1}{2}\tan\frac{\theta}{2^{n+1}}}) - \cot\theta
=\frac{1}{2^{n+1}}\cot\frac{\theta}{2^{n+1}}} - \cot\theta
\displaystyle \int_0^{\frac{\pi}{4}} ln (1+tan\theta) \hspace5 d\theta
\displaystyle -\int_{\frac{\pi}{4}}^0 ln (1 + tan (\frac{\pi}{4} - \phi) ) \hspace5 d\phi
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