Question

Show that cos 55°+cos65°+cos175°=0

Answer 1

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Answer 2

$\underline{\bold{Solution:-}}$

$LHS \\ \\ = \cos(55) + \cos(65) + \cos(175) \\ \\ Using \: identity \\ \\\bold{ \cos(x) + \cos(y) = 2 \cos( \frac{x + y}{2} ) \cos( \frac{x - y}{2} )} \\ \\ = 2 \cos( \frac{ 55 + 65}{2} ) \cos( \frac{55 - 65}{2} ) + \cos(175) \\ \\ = 2 \cos( \frac{120}{2} ) \cos( \frac{ - 10}{2} ) + \cos(175) \\ \\ = 2 \cos(60 ) \cos( - 5) + \cos(180 - 5) \\ \\ Since \: \bold{\cos( - x) = \cos(x) \: and \: 180 = \pi }\\ \\ = 2 \cos(60) \cos(5) + \cos(\pi - 5) \\ \\Since \bold{\cos(\pi - x) = - \cos(x)} \\ \\ = 2 \cos(60) \cos(5) - \cos(5)$
$\\ We \: know \: that \\ \\ \bold{\cos(60) = \frac{1}{2}} \\ \\ = 2 \times \frac{1}{2} \times \cos(5) - \cos(5) \\ \\ = \cos(5) - \cos(5) \\ \\ \boxed{= o} \\ \\ = RHS$