5. If the function $f(x)=\frac{a+\sin x}{2+\cos x}+b \tan x$ has a sum of its maximum and minimum values equal to 4, then $a+b=$ $\qquad$ .
5.3.

Given $f(x)=\frac{a+\sin x}{2+\cos x}+b \tan x$ has a maximum or minimum value, we know $b=0$.
Then $y=\frac{a+\sin x}{2+\cos x}$
$\Rightarrow \sin x-y \cos x=a-2 y$
$\Rightarrow \sin (x+\alpha)=\frac{a-2 y}{\sqrt{1+y^{2}}}$
$\Rightarrow|a-2 y| \leqslant \sqrt{1+y^{2}}$.
By Vieta's formulas, $\frac{4 a}{3}=4 \Leftrightarrow a=3$.
Thus, $a+b=3$.