Question

Prove that :

\[ \sin\frac{8\pi}{3}\cos\frac{23\pi}{6} + \cos\frac{13\pi}{3}\sin\frac{35\pi}{6} = \frac12 \]

Solution

\[ \begin{aligned} &\sin\frac{8\pi}{3}\cos\frac{23\pi}{6} + \cos\frac{13\pi}{3}\sin\frac{35\pi}{6} \\[4pt] =& \sin\frac{2\pi}{3}\cos\frac{\pi}{6} + \cos\frac{\pi}{3}\sin\left(-\frac{\pi}{6}\right) \\[4pt] =& \left(\frac{\sqrt3}{2}\times\frac{\sqrt3}{2}\right) + \left(\frac12\times-\frac12\right) \\[4pt] =& \frac34-\frac14 \\[4pt] =& \frac12 \end{aligned} \]

Hence Proved.