Step 1: Use Complex Numbers Approach
Let \( a = e^{i\alpha}, b = e^{i\beta}, c = e^{i\gamma} \).
Given condition:
\[ \sum \cos \alpha = 0 \quad \text{and} \quad \sum \sin \alpha = 0 \]
This implies \( a + b + c = (\cos \alpha + \cos \beta + \cos \gamma) + i(\sin \alpha + \sin \beta + \sin \gamma) = 0 \).
Step 2: Square the Sum
Since \( a + b + c = 0 \), squaring both sides gives:
\[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) = 0 \]
\[ a^2 + b^2 + c^2 = -2(ab + bc + ca) \]
Also, since \( |a|=|b|=|c|=1 \), we have \( \frac{1}{a} = \bar{a} \), etc.
Conjugating \( a+b+c=0 \) gives \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0 \), which implies \( ab+bc+ca = 0 \).
Thus, \( a^2 + b^2 + c^2 = 0 \).
Step 3: Analyze Imaginary Parts
\( a^2 + b^2 + c^2 = e^{i2\alpha} + e^{i2\beta} + e^{i2\gamma} = (\cos 2\alpha + \dots) + i(\sin 2\alpha + \sin 2\beta + \sin 2\gamma) = 0 \).
Comparing imaginary parts, we get:
\[ \sin 2\alpha + \sin 2\beta + \sin 2\gamma = 0 \]
Step 4: Check Options
We need to find which option equals 0.
Option (A): \( S = \cos(\alpha+\beta) + \cos(\beta+\gamma) + \cos(\gamma+\alpha) \).
Using \( a+b+c=0 \implies a+b=-c \).
Squaring modulus: \( |a+b|^2 = |c|^2 = 1 \implies 1+1+2\cos(\alpha-\beta)=1 \implies \cos(\alpha-\beta)=-1/2 \).
This implies the angles differ by \( 120^\circ \) (equilateral triangle structure).
Let \( \alpha = 0, \beta = 120^\circ, \gamma = 240^\circ \).
Target Value: \( \sin 0 + \sin 240 + \sin 480 = 0 - \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = 0 \).
Option (A): \( \cos(120) + \cos(360) + \cos(240) = -0.5 + 1 - 0.5 = 0 \).
Thus, Option (A) is the correct expression equal to the target value.