Without using the calculator, prove that: \(\sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ\)
L.H.S. = \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
R.H.S. = \(2 \sin 30^\circ \cos 30^\circ = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\)
Therefore, L.H.S. = R.H.S. = \(\frac{\sqrt{3}}{2}\)
R.H.S. = \(2 \sin 30^\circ \cos 30^\circ = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\)
Therefore, L.H.S. = R.H.S. = \(\frac{\sqrt{3}}{2}\)
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