Prove that : 13 3 2 3 + 4 3 13 6 =1 2 - Vidyadip

Question

Prove that $:\sin\frac{13\pi}{3}\sin\frac{2\pi}{3}+\cos\frac{4\pi}{3}\sin\frac{13\pi}{6}=\frac{1}{2}$

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Answer

$\text{L.H.S}=\sin\frac{13\pi}{3}\sin\frac{2\pi}{3}+\cos\frac{4\pi}{3}\sin\frac{13\pi}{6}$
$=\sin780^\circ\times\sin120^\circ+\sin240^\circ\sin390^\circ$
$=\sin\Big(4\pi+\frac{\pi}{3}\Big)\sin\Big(\frac{\pi}{2}+\frac{\pi}{6}\Big)+\cos\Big(\pi+\frac{\pi}{6}\Big)\sin\Big(2\pi+\frac{\pi}{6}\Big)$
$(\because\pi=180^\circ)$
$=\sin\frac{\pi}{3}\times\cos\frac{\pi}{6}-\cos\frac{\pi}{3}\times\Big(+\sin\frac{\pi}{6}\Big)$
$\left(\begin{array}{c}\because\sin\Big(4\pi+\frac{\pi}{3}\Big)=\sin\frac{\pi}{3}\\\&\sin\Big(3\pi-\frac{\pi}{3}\Big)=\sin\frac{\pi}{3}\end{array}\right)$
$=\frac{\sqrt{3}}{2}\times\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{1}{2}$
$=\frac{3}{4}-\frac{1}{4}$
$=\frac{1}{2}$ $=\text{R.H.S}$ $\text{Proved}$

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