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Question
The value of cos12° + cos84° + cos156° + cos132° is ______.
Options
`1/2`
1
`-1/2`
`1/8`
Solution
The value of cos12° + cos84° + cos156° + cos132° is `-1/2`.
Explanation:
The given expression is cos12° + cos84° + cos156° + cos132°
(cos132° + cos12°) + (cos156° + cos84°)
= `(2cos (132^circ + 12^circ)/2 . cos (132^circ - 12^circ)/2) + (2cos (156^circ + 84^circ)/2 . cos (156^circ - 84^circ)/2)`
= 2cos72° . cos60° + 2cos120° . cos36°
= `2 cos 72^circ xx 1/2 + 2 xx (-1/2) cos 36^circ`
= cos72° – cos36°
= cos(90° – 18°) – cos36°
= sin18° – cos36°
= `(sqrt(5) - 1)/4 - (sqrt(5) + 1)/4` ......`[because sin18^circ = (sqrt(5) - 1)/4, cos 36^circ = (sqrt(5) + 1)/4]`
= `(sqrt(5) - 1 - sqrt(5) - 1)/4`
= `-1/2`
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