Question

Prove that :

\[ \sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} = \frac12 \]

Solution

\[ \begin{aligned} &\sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} \\[8pt] =& \sin\frac{\pi}{3}\sin\frac{2\pi}{3} + \left(-\frac12\right)\left(\frac12\right) \\[8pt] =& \left(\frac{\sqrt3}{2}\right)\left(\frac{\sqrt3}{2}\right) -\frac14 \\[8pt] =& \frac34-\frac14 \\[8pt] =& \frac12 \end{aligned} \]

Hence Proved.