11.2. A triplet of real numbers $A, B, C$ is such that $\operatorname{Sin} A+\operatorname{Sin} B+\operatorname{Sin} C=0$ and $\cos A+\operatorname{Cos} B+\operatorname{Cos} C=0$. Find the value of the expression

$\operatorname{Cos}(A-B)+\operatorname{Cos}(B-C)+\operatorname{Cos}(C-A)$.
Answer: $-\frac{3}{2}$.

Solution. Squaring the first two expressions, we get $(\operatorname{Sin} A+\operatorname{Sin} B+\operatorname{Sin} C)^{2}=\operatorname{Sin}^{2} A+\operatorname{Sin}^{2} B+\operatorname{Sin}^{2} C+2(\operatorname{Sin} A \cdot \operatorname{Sin} B+\operatorname{Sin} B \cdot \operatorname{Sin} C+\operatorname{Sin} A \cdot \operatorname{Sin} C)=0$
and $(\operatorname{Cos} A+\operatorname{Cos} B+\operatorname{Cos} C)^{2}=\operatorname{Cos}^{2} A+\operatorname{Cos}^{2} B+\operatorname{Cos}^{2} C+$

$+2(\operatorname{Cos} A \cdot \operatorname{Cos} B+\operatorname{Cos} B \cdot \operatorname{Cos} C+\operatorname{Cos} A \cdot \operatorname{Cos} C)=0$. Adding the obtained expressions, we have

$3+2(\operatorname{Sin} A \cdot \operatorname{Sin} B+\operatorname{Sin} B \cdot \operatorname{Sin} C+\operatorname{Sin} A \cdot \operatorname{Sin} C+\operatorname{Cos} A \cdot \operatorname{Cos} B+\operatorname{Cos} B \cdot \operatorname{Cos} C+\operatorname{Cos} A \cdot \operatorname{Cos} C)=0$.

Considering

equalities

$\operatorname{Sin} A \cdot \operatorname{Sin} B+\operatorname{Cos} A \cdot \operatorname{Cos} B=\operatorname{Cos}(A-B), \operatorname{Sin} A \cdot \operatorname{Sin} C+\operatorname{Cos} A \cdot \operatorname{Cos} C=\operatorname{Cos}(C-A)$, and

$\operatorname{Sin} B \cdot \operatorname{Sin} C+\operatorname{Cos} B \cdot \operatorname{Cos} C=\operatorname{Cos}(B-C)$, we get

$3+2(\operatorname{Cos}(A-B)+\operatorname{Cos}(B-C)+\operatorname{Cos}(C-A))=0$, from which

$\operatorname{Cos}(A-B)+\operatorname{Cos}(B-C)+\operatorname{Cos}(C-A)=-\frac{3}{2}$.

Grading criteria. ($\cdot$) Squaring the equalities from the condition: 2 points. ($\cdot$) Adding the obtained expressions: 1 point. ($\cdot$) Using the cosine difference formulas: 3 points.