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Question
Prove that:
sin 23° + sin 37° = cos 7°
Solution
Consider LHS:
\[sin 23^\circ + \sin 37^\circ\]
\[ = 2\sin \left( \frac{23^\circ + 37^\circ}{2} \right) \cos \left( \frac{23^\circ - 37^\circ}{2} \right) \left\{ \because \sin A + \sin B = 2sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin 30^\circ \cos \left( - 7^\circ \right)\]
\[ = 2\sin 30^\circ\cos 7^\circ\]
\[ = 2 \times \frac{1}{2}\cos 7^\circ\]
\[ = \cos 7^\circ\]
Hence, LHS = RHS.
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