Step 1: Find $\sin^2 \theta$

Given the equation $4 \sin^2 \theta = 1$, we divide both sides by $4$ to find $\sin^2 \theta$.

$\sin^2 \theta = \frac{1}{4}$

Step 2: Determine $\cos^2 \theta$

Using the trigonometric identity $\cos^2 \theta = 1 - \sin^2 \theta$, we substitute the value of $\sin^2 \theta$.

$\cos^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}$

Step 3: Determine $\tan^2 \theta$

Since $\sin \theta = \frac{1}{2}$ and $\cos \theta = \frac{\sqrt{3}}{2}$ for the acute angle $\theta = 30°$, we calculate $\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}$.

$\tan^2 \theta = \frac{1/4}{3/4} = \frac{1}{3}$

Step 4: Evaluate the Required Expression

Add $\cos^2 \theta$ and $\tan^2 \theta$ by finding a common denominator ($12$).

$\cos^2 \theta + \tan^2 \theta = \frac{3}{4} + \frac{1}{3} = \frac{9 + 4}{12} = \frac{13}{12}$