Ratio of cosine sums to sine sums

Geometry Level 4

\[ \sin^{2}x_{1}+\sin^{2}x_{2}+...+\sin^{2}x_{10}=1\]

Let \(x_{1},x_{2}.....x_{10}\) be real numbers in the interval \([0,\frac{\pi}{2}]\) such that the equation above is satisfied.

If \(\frac{\cos x_{1}+\cos x_{2}+....+\cos x_{10}}{\sin x_{1}+\sin x_{2}+....+\sin x_{10}} \ge \alpha\) then find maximum value of \(\alpha\).

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