Chapter 7 - Section 7.5 - More Trigonometric Equations - 7.5 Exercises - Page 575: 45

$0$

$\cos 2\theta+\cos \theta=2$ $2\cos^2 \theta-1+\cos \theta=2$ $2\cos^2 \theta+\cos \theta-3=0$ $(2\cos \theta+3)(\cos \theta-1)=0$ If $2\cos \theta+3=0$, then $2\cos \theta=-3$, and $\cos \theta=-\frac{3}{2}$, which is impossible because $\cos \theta$ cannot be less than -1. If $\cos \theta-1=0$, then $\cos \theta=1$, and the only solution in $[0, 2\pi)$ is $\theta=0$.