<s>[INST] Maximum value of trigonometric expression If 
$r=3+\tan c \tan a, \quad  
 q=5+\tan b \tan c, \quad
 p=7+\tan a \tan b$
Provided $a,b,c$ are positive and 
$a+b+c=\dfrac{\pi}2$
Find the maximum value of 
$\sqrt p + \sqrt q + \sqrt r$ . 
 [/INST] Let $x = \tan a \tan b, \; y = \tan b \tan c, \; z = \tan c \tan a$, then $x, y, z > 0$ and $x+y+z=1$.
We need to now maximize $\sqrt{3+x}+\sqrt{5+y}+\sqrt{7+z}$.  As $\sqrt t$ is concave, using Karamata's inequality and $(3+x, 5+y, 7+z) \succ (3+x+y+z, 5, 7) = (4, 5, 7)$, we have $\sqrt{3+x}+\sqrt{5+y}+\sqrt{7+z} \le 2+\sqrt5 + \sqrt7$.
Equality is never achieved, but one can get as close to the value when $b \ll   a \to 0^+, c \to \frac{\pi}2^-$, so we do not have a maximum but a supremum.
</s>