Answer
$\frac{\pi}{12}$, $\frac{5\pi}{12}$, $\frac{7\pi}{12}$, $\frac{11\pi}{12}$, $\frac{13\pi}{12}$, $\frac{17\pi}{12}$, $\frac{19\pi}{12}$, $\frac{23\pi}{12}$
Work Step by Step
a. $\sec 4\theta-2=0$
$\sec 4\theta=2$
$\cos4\theta=\frac{1}{2}$
$4\theta=\frac{\pi}{3}+2k\pi$, $\frac{5\pi}{3}+2k\pi$
$\theta=\frac{\pi}{12}+\frac{k\pi}{2}$, $\frac{5\pi}{12}+\frac{k\pi}{2}$
b. If $\theta=\frac{\pi}{12}+\frac{k\pi}{2}$, the only solutions in $[0, 2\pi)$ are $\frac{\pi}{12}$, $\frac{7\pi}{12}$, $\frac{13\pi}{12}$ and $\frac{19\pi}{12}$. If $\theta=\frac{5\pi}{12}+\frac{k\pi}{2}$, the only solutions in $[0, 2\pi)$ are $\frac{5\pi}{12}$, $\frac{11\pi}{12}$, $\frac{17\pi}{12}$ and $\frac{23\pi}{12}$.
