Prove that $\cos 55 ° + \cos 65 ° + \cos 175 ° = 0$ .

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Solution

$L . H . S = \cos 55 ° + \cos 65 ° + \cos 175 °$
$= 2 . \cos (\frac{55 ° + 65 °}{2}) \cos (\frac{55 ° - 65 °}{2}) + \cos 175 °$ [ Since, $\cos A + \cos B = 2 \cos (\frac{A + B}{2}) \cos (\frac{A - B}{2})$ ]
$= 2 . \cos 60 ° \cos (- 5 °) + \cos 175 °$
$= 2 x \frac{1}{2} \cos 5 ° + \cos 175 °$
$= \cos 5 ° + \cos 175 °$
$= 2 \cos (\frac{5 ° + 175 °}{2}) \cos (\frac{5 ° - 175 °}{2})$
$= 2 \cos 90 ° \cos 85 °$
$= 2 (0) \cos 85 °$
$= 0$
$= R . H . S .$
Therefore, $L H S = R H S$
Hence, it is proved that $\cos 55 ° + \cos 65 ° + \cos 175 ° = 0$

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