<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 $\sqrt p=P$ etc.
$$R^2-3+Q^2-5+P^2-7=1\iff P^2+Q^2+R^2=16$$
Let $P=4\cos A,Q=4\sin A\cos B,R=4\sin A\sin B$
$P+Q+R=4[\cos A+\sin A(\cos B+\sin B)]\le4[\cos A+\sqrt2\sin A]$ if $\sin A\ge0$
Now $\cos A+\sqrt2\sin A=\sqrt3\cos\left(A-\arccos\dfrac1{\sqrt3}\right)\le\sqrt3$
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