Sine possibility

$\begin{aligned} \left(1+\frac 1{\sin x}\right)\cos x + \left(1+\frac 1{\cos x}\right)\sin x & = 2 + \sqrt 2 \\ \cos x + \cot x + \sin x + \tan x & = 2 + \sqrt 2 \\ \sin x + \cos x - \sqrt 2 & = 2 - \tan x - \cot x \\ \sqrt 2 \sin \left(x + \frac \pi 4 \right) - \sqrt 2 & = 2 - (\tan x + \cot x) \end{aligned}$

Note that $-\sqrt 2 \le \sqrt 2 \sin \left(x + \frac \pi 4 \right) \le \sqrt 2$ then the left-hand side is $\rm -2\sqrt 2 \le LHS \le 0$ . For $0 \le x < 2\pi$ $\rm LHS = 0$ , when $x = \frac \pi 4$ .

Considering $\tan x + \cot x > 0$ , by AM-GM inequality $\tan x + \cot x \ge 2 \sqrt{\tan x \cot x} = 2$ . Then the right-hand side, $\rm RHS \le 0$ . For $0\le x < 2\pi$ , $\rm RHS = 0$ , when $x = \frac \pi 4$ and $x= \frac 54 \pi$ . Therefore, $\rm LHS = RHS$ , when $x = \frac \pi 4$ and a value of $\sin x = \sin \frac \pi 4 = \frac 1{\sqrt 2}$ . Since the $\rm RHS = -2\sqrt 2$ , when $x \approx 0.214, 1.357, 3.355, 4.499$ , there are two other points when $\rm LHS = RHS$ . Therefore there are $\boxed 3$ possible values of $\sin x$ .