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Question
Prove that:
cos 55° + cos 65° + cos 175° = 0
Solution
Consider LHS:
\[\cos 55^\circ + \cos 65^\circ + \cos 175^\circ\]
\[ = 2\cos \left( \frac{55^\circ + 65^\circ}{2} \right) \cos \left( \frac{55^\circ - 65^\circ}{2} \right) + \cos 175^\circ \left\{ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 60^\circ \cos\left( - 5^\circ \right) + \cos 175^\circ\]
\[ = 2 \times \frac{1}{2}\cos 5^\circ + \cos 175^\circ\]
\[ = \cos 5^\circ + \cos 175^\circ\]
\[ = 2\cos \left( \frac{5^\circ + 175^\circ}{2} \right) \cos \left( \frac{5^\circ - 175^\circ}{2} \right)\]
\[ = 2\cos 90^\circ \cos 85^\circ\]
\[ = 0\]
Hence, LHS = RHS.
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