How to prove $\prod_{n=1} ^{22} \frac{\tan(2n-1)}{\tan(2n)} \lt \sqrt{\tan(1)}$

Question

How to prove $$P = \prod_{n=1} ^{22} \frac{\tan(2n-1)}{\tan(2n)} \lt \sqrt{\tan(1)}$$ I reached the following $$P^2 = \tan(1) \prod_{n=1} ^{22} \frac{\tan(2n-1) \tan(2n+1)}{{\tan(2n)}^2}$$ But don't know how to proceed.

Answer 1

It's wrong! $$\prod_{k=1}^{22}\frac{\tan(2k-1)}{\tan2k}=7377.3...>\sqrt{\tan1}.$$ See here: http://www.wolframalpha.com/input/?i=prod_%7Bk%3D1%7D%5E22(tan(2k-1)%2Ftan(2k))