Question

The value of $cos12^{\circ }+cos84^{\circ }+cos156^{\circ }+cos132^{\circ }$ is

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

Answer 1

Answer: (C)

$cos12^{\circ }+cos84^{\circ }+cos156^{\circ }+cos132^{\circ }$
$=cos156^{\circ }+cos84^{\circ }+cos132^{\circ }+cos12^{\circ }$
$=2cos\left(\frac{156^{\circ }+84^{\circ }}{2}\right)cos\left(\frac{156^{\circ }-84^{\circ }}{2}\right)$
$+2cos\left(\frac{132^{\circ }+12^{\circ }}{2}\right)cos\left(\frac{132^{\circ }-12^{\circ }}{2}\right)$
$=2cos120^{\circ }cos36^{\circ }+2cos72^{\circ }cos60^{\circ }$
$=2\left(-\frac{1}{2}\right)cos36^{\circ }+2cos72^{\circ }\times \frac{1}{2}$
$=-cos36^{\circ }+cos72^{\circ }$
$=-\frac{\sqrt{5}+1}{4}+\frac{\sqrt{5}-1}{4}$
$=-\frac{1}{2}$