Question

Solve:
$\cos {12}^o+\cos {84}^o+\cos {156}^o+\cos {132}^o$

Answer 1

$\cos {12}^{\circ }+\cos {84}^{\circ }+\cos {156}^{\circ }+\cos {132}^{\circ }=2\cos {48}^{\circ }\cos {36}^{\circ }+2\cos {144}^{\circ }\cos {12}^{\circ }=2\cos {48}^{\circ }\cos {36}^{\circ }-2\cos {36}^{\circ }\cos {12}^{\circ }$

$=2\cos {36}^{\circ }(\cos {48}^{\circ }-\cos {12}^{\circ })=-4\cos {36}^{\circ }\sin {30}^{\circ }\sin {18}^{\circ }=-2\times (\frac{\sqrt{5}-1}{4})(\frac{\sqrt{5}+1}{4})=-\frac{1}{2}$