Step-by-Step Solution
Key Concept: Clear denominators in trigonometric equations by multiplying strategically, then solve the resulting polynomial equation
Starting with $\tan x + \sec x = 2\cos x$, multiply both sides by $\cos x$ to get $\sin x + 1 = 2\cos^2 x = 2(1 - \sin^2 x)$. This simplifies to $2\sin^2 x + \sin x - 1 = 0$. Factoring: $(2\sin x - 1)(\sin x + 1) = 0$ gives $\sin x = \frac{1}{2}$ or $\sin x = -1$. The solution $\sin x = -1$ is extraneous (makes original equation undefined). For $\sin x = \frac{1}{2}$, we have $x = \frac{\pi}{6}$ or $x = \frac{5\pi}{6}$ in $[0, 2\pi)$. Only one valid solution remains after verification.
Correct Answer: 1