Question

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

• A. $\frac{1}{2}$
• B. $1$
• C. $-\frac{1}{2}$
• D. $\frac{1}{8}$

Answer 1

Answer: (C)

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