# Ratio of two tanned alternate signed even-odd binomial summation

**Geometry**Level 5

\[\large R=\dfrac{\displaystyle\sum_{r=1}^{2k} (-1)^{r-1}\binom{4k}{2r-1}\tan^{2r-1}\theta}{\displaystyle\sum_{r=0}^{2k} (-1)^{r}\binom{4k}{2r}\tan^{2r}\theta}\]

Find the value of \(R\) if \( k=100\) , \(\theta = \dfrac{\pi}{480}\).

Give your answer to correct 3 decimal places.