I'm not quite sure if I'm answering the question...?

"Use the double argument property, $\displaystyle cos 2x = 1 - 2 sin2x$ to express cos 60° in terms of sin 30°."

My response:

$\displaystyle x=30°

cos(2*30°)=1-2sin^2(30°)

cos(60°)=1-2sin^2(30°)

(cos(60°)-1)/-2)=sin^2(30°)

0.25=sin^2(30°)

0.5=sin(30°)$