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Question
Solution
LHS =\[ \cos \left( 570^\circ \right)\sin \left( 510^\circ \right) + \sin \left( - 330^\circ \right)\cos \left( - 390^\circ \right)\]
\[ = \cos \left( 570^\circ \right) \sin \left( 510^\circ \right) + \left[ - \sin \left( 330^\circ \right) \right]\cos \left( 390^\circ \right) \left[ \because \sin\left( - x \right) = - \sin x and \cos\left( - x \right) = \cos x \right] \]
\[ = \cos \left( 570^\circ \right)\sin\left( 510^\circ \right) - \sin \left( 330^\circ\right)\]
\[ = \cos \left( 90^\circ \times 6 + 30^\circ \right) \sin \left( 90^\circ \times 5 + 60^\circ \right) - \sin \left( 90^\circ \times 3 + 60^\circ \right) \cos \left( 90^\circ \times 4 + 30^\circ \right)\]
\[ = - \cos \left( 30^\circ \right) \cos \left( 60^\circ \right) - \left[ - \cos \left( 60^\circ \right) \right] \cos \left( 30^\circ \right)\]
\[ = - \cos \left( 30^\circ \right) \cos \left( 60^\circ \right) + \cos \left( 30^\circ \right) \sin \left( 60^\circ \right)\]
\[ = 0\]
= RHS
Hence proved .
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