Q.If $x + iy = \operatorname{cis}\alpha \cdot \operatorname{cis}\beta$, then find the value of $x^2 + y^2$.

$\operatorname{cis}\alpha\cdot\operatorname{cis}\beta$ is itself of the form $\cos\theta+i\sin\theta$, whose modulus is always $1$, so $x^2+y^2=1$.

Recall $\operatorname{cis}\theta = \cos\theta + i\sin\theta$. Using the product rule for $\operatorname{cis}$ (which follows from the compound-angle formulas):

$\operatorname{cis}\alpha\cdot\operatorname{cis}\beta = (\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta) = \cos(\alpha+\beta) + i\sin(\alpha+\beta) = \operatorname{cis}(\alpha+\beta)$ …