Questions: Use the sum-to-product identities to rewrite the following expression as a product. cos (5π/6) - cos (π/6)

Solution Steps

We start with the expression: \[ \cos \left(\frac{5 \pi}{6}\right) - \cos \left(\frac{\pi}{6}\right) \] Using the sum-to-product identity: \[ \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \] we set \(A = \frac{5\pi}{6}\) and \(B = \frac{\pi}{6}\).

Calculating \(A + B\) and \(A - B\): \[ A + B = \frac{5\pi}{6} + \frac{\pi}{6} = \frac{6\pi}{6} = \pi \] \[ A - B = \frac{5\pi}{6} - \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

Substituting these values into the identity gives: \[ \cos \left(\frac{5 \pi}{6}\right) - \cos \left(\frac{\pi}{6}\right) = -2 \sin \left(\frac{\pi}{2}\right) \sin \left(\frac{2\pi}{3}\right) \]

We know: \[ \sin \left(\frac{\pi}{2}\right) = 1 \] and \[ \sin \left(\frac{2\pi}{3}\right) = \sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \] Thus, we have: \[ \cos \left(\frac{5 \pi}{6}\right) - \cos \left(\frac{\pi}{6}\right) = -2 \cdot 1 \cdot \frac{\sqrt{3}}{2} = -\sqrt{3} \]

Final Answer