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Mathematics Question on Trigonometric Functions

The value of cos 12∘+cos84∘+cos 156∘+cos 132∘cos \,12^{\circ} + cos84^{\circ} + cos \,156^{\circ} + cos \,132^{\circ} is

A

12\frac{1}{2}

B

11

C

−12-\frac{1}{2}

D

18\frac{1}{8}

Answer

−12-\frac{1}{2}

Explanation

Solution

cos 12∘+cos84∘+cos 156∘+cos 132∘cos \,12^{\circ} + cos84^{\circ} + cos \,156^{\circ} + cos \,132^{\circ} =cos 156∘+cos84∘+cos 132∘+cos 12∘=cos \,156^{\circ} + cos84^{\circ} + cos \,132^{\circ} + cos \,12^{\circ} =2 cos(156∘+84∘2)cos(156∘−84∘2)=2\,cos\left(\frac{156^{\circ}+84^{\circ}}{2}\right)cos\left(\frac{156^{\circ}-84^{\circ}}{2}\right) +2 cos(132∘+12∘2)cos(132∘−12∘2)+2\,cos\left(\frac{132^{\circ}+12^{\circ}}{2}\right)cos\left(\frac{132^{\circ}-12^{\circ}}{2}\right) =2 cos 120∘ cos36∘+2cos72∘ cos60∘=2\,cos\,120^{\circ}\,cos36^{\circ}+2cos72^{\circ}\,cos60^{\circ} =2(−12)cos36∘+2cos72∘×12=2\left(-\frac{1}{2}\right)cos36^{\circ}+2cos72^{\circ}\times\frac{1}{2} =−cos36∘+cos72∘=-cos36^{\circ}+cos72^{\circ} =−5+14+5−14=-\frac{\sqrt{5}+1}{4}+\frac{\sqrt{5}-1}{4} =−12=-\frac{1}{2}