Question

Prove that:
sin 40° + sin 20° = cos 10°

Answer 1

Consider LHS: 
\[\sin 40^\circ + \sin 20^\circ\]
\[ = 2\sin \left( \frac{40^\circ + 20^\circ}{2} \right) \cos \left( \frac{40^\circ - 20^\circ}{2} \right) \left\{ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin 30^\circ \cos 10^\circ\]
\[ = 2 \times \frac{1}{2}\cos 10^\circ\]
\[ = \cos\left( 10^\circ \right)\]
Hence, LHS = RHS.