Let $a,b,c\in\mathbb{R}$ s.t $\cos(a-b) + \cos(b-c) + \cos(c-a) = -3/2$.
Find $$\sin(a+b) + \sin(b+c) + \sin(c+a)$$
My work: I begin with writing $\sin(a+b)$ as $\dfrac{e^{i(a+b)} - e^{-i(a+b)}}{2i}$ and rest in similar way, But cannot simplify further.
Image of question: https://i.stack.imgur.com/8g1sB.jpg
You have $\exp(i(a-b))+\exp(i(b-a))+\exp(i(b-c))+\exp(i(c-b))+\exp(i(c-a))+\exp(i(a-c))=-3$. Now compute $(\exp(ia)+\exp(ib)+\exp(ic))(\exp(-ia)+\exp(-ib)+\exp(-ic))$, one find $3-3=0$. This gives $\exp(ia)+\exp(ib)+\exp(ic)=0$, and multiplying by $\exp(-i(a+b+c))$ and taking real and imaginary parts you have the result.