Minimum value

[man111]

If $\displaystyle x_{i}\in \left[0,\frac{\pi}{2}\right]\;\forall i=1,2,3,4,....,10$. and $\sin^2x_{1}+\sin^2x_{2}+....+\sin^2x_{10} = 1$

Then Minimum value of $\displaystyle \frac{\cos x_{1}+\cos x_{2}+......+\cos x_{10}}{\sin x_{1}+\sin x_{2}+....+\sin x_{10}}$

My attempt:: Using $\sin^2 x_{i}+\cos^2 x_{i} = 1\;,$ We get $\cos^2x_{1}+\cos^2x_{2}+....+\cos^2x_{10} = 9$

And using $\displaystyle \cos^2 x_{i}\leq \cos x_{i}\;\forall \cos x_{i}\in \left[0,1\right]$

So $\cos x_{1}+\cos x_{2}+......+\cos x_{10}\geq 9$

Now how can i calculate after that, Help me

Thanks

[Sergic Primazon]

$\frac {\sqrt {1-x_1}+...\sqrt {1-x_n}}{\sqrt x_1+...+\sqrt x_n} \ge \sqrt {n-1}, x_1+...+x_n=1$

$f (t)= \sqrt {1-t}- \sqrt {n-1}\cdot \sqrt {t} \ge -\frac {n^{3/2}}{2\sqrt {n-1}}(t-\frac {1}{n})$ при $n \ge 4$

[man111]

Thanks Sergic Primazon, Would you like to explain me