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[tex] \cos( \frac{5\pi}{12} ) = \cos( \frac{6\pi}{12} - \frac{\pi}{12} ) = \\ [/tex]
[tex] \cos( \frac{\pi}{2} - \frac{\pi}{12} ) = \sin( \frac{\pi}{12} ) \\ [/tex]
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Also we have :
[tex] \sin( \frac{5\pi}{12} ) = \cos( \frac{\pi}{12} ) \\ [/tex]
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So :
[tex] \cos( \frac{\pi}{12} ). \sin( \frac{\pi}{12} ) + \sin( \frac{\pi}{12} ) . \cos( \frac{\pi}{12} ) = \\ [/tex]
[tex]2 \sin( \frac{\pi}{12} ) . \cos( \frac{\pi}{12} ) \\ [/tex]
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Hint :
[tex] \sin(2 \alpha ) = 2 \sin( \alpha ) . \cos( \alpha ) [/tex]
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So ;
[tex]2 \sin( \frac{\pi}{12} ) . \cos( \frac{\pi}{12} ) = \\ [/tex]
[tex] \sin(2 \times \frac{\pi}{12} ) = \sin( \frac{\pi}{6} ) \\ [/tex]
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[tex] \sin( \frac{\pi}{6} ) = \cos( - \frac{\pi}{3} ) \\ [/tex]
Thus the correct answer is the first option.
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