Answer
A complex number in trigonometric form can be squared by means of FOIL expansion and the use of Double-Angle Identities / Sum Identities for Cosine and Sine.
Work Step by Step
$(rcis\theta)^2$ = $(rcis\theta)(rcis\theta)$ = $r^2cis(\theta + \theta)$ = $r^2cis2\theta$ (as given)
$(rcis\theta)^2$
= $r^2(cis\theta)^2$
= $r^2(cos\theta + isin\theta)^2$
= $r^2(cos^2\theta + 2isin\theta cos\theta + i^2sin^2\theta)$
= $r^2(cos^2\theta - sin^2\theta + 2sin\theta cos\theta i)$ $(i^2 = -1)$
= $r^2(cos2\theta + 2sin\theta cos\theta i)$ $(cos^2\theta - sin^2\theta = cos2\theta$ since Double-Angle Identities / Sum Identities for Cosine$)$
= $r^2(cos2\theta + i sin2\theta)$ $(2sin\theta cos\theta = sin2\theta$ since Double-Angle Identities$)$
= $r^2cos2\theta$
