Answer
$\dfrac{1}{4}$
Work Step by Step
Let us consider:
$g(\theta)=\cos \theta$
We are given that $\theta=60^{\circ}$
We know form the unit circle that
$\cos (60^{\circ})=\cos (\dfrac{\pi}{3})=\dfrac{1}{2}$
Thus, we have: $[g(\theta)]^2=[cos(60^{\circ})]^2
\\=[cos\dfrac{\pi}{3}]^2 \\ =[\dfrac{1}{2}]^2 \\=\dfrac{1}{4}$