Question:

Step 1: Evaluate $f\left(\dfrac{\pi}{7}\right)$ by substituting $\theta = \dfrac{\pi}{7}$. Step 2: Compute $\sin 6\theta = \sin\dfrac{6\pi}{7} = \sin\dfrac{\pi}{7}$ and $\sin 5\theta = \sin\dfrac{5\pi}{7} = \sin\dfrac{2\pi}{7}$. Step 3: First factor: $1 + \dfrac{4\sin(\pi/7)}{\sin(6\pi/7)} = 1 + \dfrac{4\sin(\pi/7)}{\sin(\pi/7)} = 1 + 4 = 5$. Step 4: Second factor: $1 + \dfrac{4\sin(2\pi/7)}{\sin(5\pi/7)} = 1 + \dfrac{4\sin(2\pi/7)}{\sin(2\pi/7)} = 1 + 4 = 5$. Step 5: Therefore $f\left(\dfrac{\pi}{7}\right) = 5 \times 5 = 25$. Option (a) is correct. Step 6: Evaluate $f\left(\dfrac{2\pi}{7}\right)$ by substituting $\theta = \dfrac{2\pi}{7}$. Step 7: Compute $\sin 6\theta = \sin\dfrac{12\pi}{7} = -\sin\dfrac{2\pi}{7}$ and $\sin 5\theta = \sin\dfrac{10\pi}{7} = -\sin\dfrac{3\pi}{7}$. Step 8: First factor: $1 + \dfrac{4\sin(2\pi/7)}{-\sin(2\pi/7)} = 1 - 4 = -3$. Step 9: Second factor: $1 + \dfrac{4\sin(4\pi/7)}{-\sin(3\pi/7)} = 1 - \dfrac{4\sin(3\pi/7)}{\sin(3\pi/7)} = 1 - 4 = -3$. Step 10: Therefore $f\left(\dfrac{2\pi}{7}\right) = (-3)\times(-3) = 9$. Option (c) is correct.