Question

Prove that:
 cos 100° + cos 20° = cos 40°

Answer 1

Consider LHS: 
\[\cos 100^\circ + \cos 20^\circ\]
\[ = 2\cos \left( \frac{100^\circ + 20^\circ}{2} \right) \cos \left( \frac{100^\circ - 20^\circ}{2} \right) \left\{ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 60^\circ \cos 40^\circ\]
\[ = 2 \times \frac{1}{2}\cos 40^\circ\]
\[ = \cos 40^\circ\]
Hence, LHS = RHS.