Answer
$\dfrac{-1}{2}$
Work Step by Step
Let us consider
$g( \theta)=\cos \theta$
When $\theta=60^{\circ}$, then $2 \theta=2 (60^{\circ})=120^{\circ}$
We know from the unit circle that $\cos (120^{\circ})=\cos (\dfrac{2\pi}{3})=\dfrac{-1}{2}$
The value of the cosine function is negative because the angle $(\dfrac{2\pi}{3})$ lies in the second quadrant.
Thus, we have: $g(2\theta)=\cos (120^{\circ})
\\=\cos (\dfrac{2\pi}{3}) \\ =\dfrac{-1}{2}$