Prove that in any triangle ABC:
$\sqrt{(\tan \frac{A}{2} + \tan \frac{B}{2})(\tan \frac{B}{2}+\tan \frac{C}{2})} + \sqrt{(\tan \frac{B}{2} + \tan \frac{C}{2})(\tan \frac{C}{2}+\tan \frac{A}{2})}+ \sqrt{(\tan \frac{C}{2} + \tan \frac{A}{2})(\tan \frac{A}{2}+\tan \frac{B}{2})} \leq 2(\cot A + \cot B + \cot C)$
@leezangqe:select your whole latex code and click the summation symbol on the bar.
we know that
lets start with the left side of the inequality:
(# as in a triangle, and using product-sum formula)
now,
so similarly we can write,
Adding these together we get,
now lets simplify the right side:
by the AM-GM inequality we can write:
or, similarly,
adding these we get,
from
proved