Question

Prove that:

sin 51° + cos 81° = cos 21°

Answer 1

Consider LHS: 
\[\sin 51^\circ + \cos 81^\circ\]
\[ = \sin 51^\circ + \cos \left( 90^\circ - 9^\circ \right)\]
\[ = \sin 51^\circ + \sin 9^\circ\]
\[ = 2\sin \left( \frac{51^\circ + 9^\circ}{2} \right) \cos \left( \frac{51^\circ - 9^\circ}{2} \right) \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin 30^\circ \cos 21^\circ\]
\[ = 2 \times \frac{1}{2}\cos\left( 21^\circ \right)\]
\[ = \cos\left( 21^\circ \right)\]
 = RHS
Hence, LHS = RHS.