Question:

Step 1: Convert the given conditions into algebraic equations. From the given ratios, we can write: $$\cos x + \cos y + \cos z = 2\cos(x+y+z) \quad \text{...(1)}$$ $$\sin x + \sin y + \sin z = 2\sin(x+y+z) \quad \text{...(2)}$$ Step 2: Combine equations (1) and (2) using complex exponentials. Multiply equation (2) by $i$ and add it to equation (1): $$(\cos x + \cos y + \cos z) + i(\sin x + \sin y + \sin z) = 2\cos(x+y+z) + 2i\sin(x+y+z)$$ Using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, the left side becomes $e^{ix} + e^{iy} + e^{iz}$ and the right side becomes $2e^{i(x+y+z)}$: $$e^{ix} + e^{iy} + e^{iz} = 2e^{i(x+y+z)} \quad \text{...(3)}$$ Step 3: Express the required sum using complex exponentials. We need to find $\cos(x+y) + \cos(y+z) + \cos(z+x)$. Using Euler's formula, this equals the real part of: $$e^{i(x+y)} + e^{i(y+z)} + e^{i(z+x)}$$ Step 4: Factor out the common exponential term. We can rewrite the sum as: $$e^{i(x+y)} + e^{i(y+z)} + e^{i(z+x)} = e^{i(x+y+z)} \cdot e^{-iz} + e^{i(x+y+z)} \cdot e^{-ix} + e^{i(x+y+z)} \cdot e^{-iy}$$ Factoring out $e^{i(x+y+z)}$: $$= e^{i(x+y+z)}(e^{-iz} + e^{-ix} + e^{-iy})$$ Step 5: Recognize the conjugate relationship. Note that $e^{-iz} + e^{-ix} + e^{-iy}$ is the complex conjugate of $e^{ix} + e^{iy} + e^{iz}$: $$e^{i(x+y)} + e^{i(y+z)} + e^{i(z+x)} = e^{i(x+y+z)} \cdot \overline{(e^{ix} + e^{iy} + e^{iz})}$$ Step 6: Substitute equation (3) into the expression. From equation (3), we know that $e^{ix} + e^{iy} + e^{iz} = 2e^{i(x+y+z)}$. Therefore: $$\overline{(e^{ix} + e^{iy} + e^{iz})} = \overline{2e^{i(x+y+z)}} = 2e^{-i(x+y+z)}$$ Substituting: $$e^{i(x+y)} + e^{i(y+z)} + e^{i(z+x)} = e^{i(x+y+z)} \cdot 2e^{-i(x+y+z)} = 2$$ Step 7: Extract the real part to find the final answer. Taking the real part of both sides: $$\cos(x+y) + \cos(y+z) + \cos(z+x) = \text{Re}(2) = 2$$ **Final Answer:** The value of $\cos(x+y) + \cos(y+z) + \cos(z+x) = 2$ This corresponds to **Option 3: 2** <div class="key-concept"><strong>Key Concept:</strong> Complex exponential representation of trigonometric sums</div> <div class="trap-box"><strong>Trap:</strong> Use complex exponentials to elegantly combine the two given conditions and evaluate the required expression.</div>