Answer
See proof
Work Step by Step
Let $f(x) = 2x+\cos x$
Then $f(-\pi)$ = $-2\pi-1$ $\lt$ $0$ and $f(0)$ = $1$ $\gt$ $0$
Since $f$ is the sum of the polynomial $2x$ and the trigonometric function $\cos x$ such that $f(c)$ = $0$. Thus the given equation has at least one real root.
If the equation has distinct real roots $a$ and $b$ with $a0$ since $\sin r \leq 1$.
This contradiction shows that the given equation can't have $2$ distinct real roots, so it has exactly one root.
