Question

Prove that:

sin 80° − cos 70° = cos 50°

Answer 1

Consider LHS: 
\[\sin 80^\circ - \cos 70^\circ\]
\[ = \sin 80^\circ - \cos \left( 90^\circ - 20^\circ \right)\]
\[ = \sin 80^\circ - \sin 20^\circ\]
\[ = 2\sin \left( \frac{80^\circ - 20^\circ}{2} \right) \cos \left( \frac{80^\circ + 20^\circ}{2} \right) \left\{ \because \sin A - \sin B = 2\sin \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right) \right\}\]
\[ = 2\sin 30^\circ \cos 50^\circ\]
\[ = 2 \times \frac{1}{2}\cos 50^\circ\]
\[ = \cos 50^\circ\]
= RHS
Hence, LHS = RHS.