Answer
a) No.
b) Yes.
Work Step by Step
a)$$\cos 60^{\circ } = \cos ( 2 \cdot 30^{\circ })=\frac{1}{2}, \quad \cos 30^{\circ }= \frac{\sqrt{3}}{2} \\ \Rightarrow \quad \cos ( 2 \cdot 30^{\circ })= \frac{1}{2} \neq \sqrt{3}=2(\frac{\sqrt{3}}{2})= 2 \cos 30^{\circ }$$
Thus, they are not equal.
b)$$\cos 60^{\circ } = \cos ( 2 \cdot 30^{\circ })=\frac{1}{2}, \quad \cos 30^{\circ }= \frac{\sqrt{3}}{2}, \quad \sin 30^{\circ }=\frac{1}{2} \\ \Rightarrow \quad \cos ( 2 \cdot 30^{\circ })= \frac{1}{2} =\frac{3}{4}-\frac{1}{4}= \cos ^2 30^{\circ } - \sin ^2 30^{\circ }$$
Thus, they are equal.
