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Question
Prove that:
cos 100° + cos 20° = cos 40°
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Solution
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.
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