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Prove without using a calculator, that $\dfrac{\sin{315^{\circ}}.\tan{210^{\circ}}.\sin{190^{\circ}}}{\cos{100^{\circ}}.\sin{120^{\circ}}} = \dfrac{-\sqrt{2}}{3}$
in Mathematics by Diamond (79,336 points) | 192 views

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LHS

\(= -\sin{45^{\circ}}.\tan{30^{\circ}}\cdot \dfrac{-\sin{10^{\circ}}}{-\cos{80}^{\circ}}\cdot \sin{60}^{\circ}\)

\(=\dfrac{ -\sqrt{2}}{2} \times  \dfrac{\sqrt{3}}{3} \times \dfrac{-\sin{10}^{\circ}}{-\sin{10}^{\circ}} \cdot \dfrac{\sqrt{3}}{2} \)

\(=\dfrac{ -\sqrt{2}}{3}\)  , proven
by Diamond (44,112 points)
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