Question

The value of $\cos {12}^o+\cos {84}^o+\cos {156}^o+\cos {132}^o$ is

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

Answer 1

The correct option is C $-\frac{1}{2}$

$\cos {12}^{\circ }+\cos {84}^{\circ }+\cos {156}^{\circ }+\cos {132}^{\circ }=(\cos {132}^{\circ }+\cos {12}^{\circ })+(\cos {84}^{\circ }+\cos {156}^{\circ })$

$=2\cos (\frac{{132}^{\circ }-{12}^{\circ }}{2})\cos (\frac{{132}^{\circ }+{12}^{\circ }}{2})+2\cos (\frac{{84}^{\circ }+{156}^{\circ }}{2})\cos (\frac{{156}^{\circ }-{84}^{\circ }}{2})$

$=2\cos {60}^{\circ }\cos {72}^{\circ }+2\cos {120}^{\circ }\cos {36}^{\circ }$

$=2\times \frac{1}{2}\times \frac{\sqrt{5}-1}{4}+2\times (-\frac{1}{2})\times \frac{\sqrt{5}+1}{4}$

$=\frac{\sqrt{5}-1}{4}-\frac{\sqrt{5}+1}{4}$

$=-\frac{1}{2}$