If $x=\cos A+i\sin A$ and $y=\cos B+i\sin B$ then the value of $\cos(A+B)$ in terms of $x$ and $ y$ is

Question

$$xy=\cos(A+B)+i\sin(A+B)$$ which is obtained by simply multiplying $x$ and $y$. What I should I do to further solve it?

Answer is $\frac 12 (xy+\frac{1}{xy})$

Answer 1

$$xy=e^{i(A+B)} $$

$$\bar{xy}=e^{-i(A+B)}= 1/{xy}$$

$$\cos(A+B) = Re (xy) = (1/2)(xy +\bar {xy}) = (1/2)(xy + 1/{xy})$$

Answer 2

$$\dfrac1{xy}=\cdots=\cos(A+B)-i\sin(A+B)$$ by rationalizing the denominator

Answer 3

I would write real(xy). As far as I am concerned, taking the real part is a perfectly acceptable operation.