Prove that: 570^ 510^ + (-330^ ) (-390^ )=0 - Vidyadip

Question

Prove that: $\cos570^\circ\sin510^\circ+\sin(-330^\circ)\cos(-390^\circ)=0$

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Answer

$\text{L.H.S}=\cos570^\circ\sin510^\circ+\sin(-330^\circ)\cos(-390^\circ)$ $\cos\Big(3\pi+\frac{\pi}{6}\Big)\sin\Big(3\pi-\frac{\pi}{6}\Big)-\sin330^\circ\cos390^\circ$$\begin{pmatrix}\because\sin(-\theta)=-\sin\theta\text{ and}\\\cos(-\theta)=\cos\theta\end{pmatrix}$ $=-\cos\frac{\pi}{6}\sin\frac{\pi}{6}-\sin\Big(2\pi-\frac{\pi}{6}\Big)\cos\Big(2\pi+\frac{\pi}{6}\Big)$ $=-\sin\frac{\pi}{6}\cos\frac{\pi}{6}+\sin\frac{\pi}{6}.\cos\frac{\pi}{6}$ $(\because\sin(2\pi-\theta)=-\sin\theta)$ $= 0$ $\text{= R.H.S}$ $\text{Proved}$

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