Question:

Step 1: Apply the power reduction formula for $\sin^2\theta$. We use the identity $\sin^2\theta = \frac{1-\cos 2\theta}{2}$ to rewrite each term in the sum: $$S = \sum_{\alpha=1}^{17} \sin^2(5\alpha)° = \sum_{\alpha=1}^{17} \frac{1-\cos(10\alpha)°}{2}$$ Step 2: Separate the sum into two parts. We can split the summation: $$S = \sum_{\alpha=1}^{17} \frac{1}{2} - \sum_{\alpha=1}^{17} \frac{\cos(10\alpha)°}{2}$$ $$S = \frac{17}{2} - \frac{1}{2}\sum_{\alpha=1}^{17}\cos(10\alpha)°$$ Step 3: Identify the sum of cosines as a geometric series. The sum $\sum_{\alpha=1}^{17}\cos(10\alpha)°$ represents cosines of angles: $10°, 20°, 30°, \ldots, 170°$. This is a sum of cosines in arithmetic progression, which can be evaluated using the formula: $$\sum_{k=1}^{n}\cos(k\theta) = \frac{\sin(n\theta/2)}{\sin(\theta/2)}\cos\left(\frac{(n+1)\theta}{2}\right)$$ Step 4: Apply the formula with $n=17$ and $\theta=10°$. Substituting our values: $$\sum_{\alpha=1}^{17}\cos(10\alpha)° = \frac{\sin(17 \cdot 10°/2)}{\sin(10°/2)}\cos\left(\frac{18 \cdot 10°}{2}\right)$$ $$= \frac{\sin(85°)}{\sin(5°)}\cos(90°)$$ Step 5: Evaluate using $\cos(90°) = 0$. Since $\cos(90°) = 0$: $$\sum_{\alpha=1}^{17}\cos(10\alpha)° = \frac{\sin(85°)}{\sin(5°)} \cdot 0 = 0$$ Step 6: Calculate the final value of $S$. Substituting back into our expression for $S$: $$S = \frac{17}{2} - \frac{1}{2}(0) = \frac{17}{2} = 8.5$$ Step 7: Apply the greatest integer function. The greatest integer function $[S]$ gives us the largest integer less than or equal to $S$: $$[S] = [8.5] = 8$$ **Final Answer:** $[S] = 8$, which corresponds to **Option 2**.