Question

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

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

Answer 1

Answer: (C)

$- \frac{1}{2}$

Answer 2

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

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

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

$= 2\left\lbrack \cos{}72^{o} \times \frac{1}{2} - \frac{1}{2} \times \cos{}36^{o} \right\rbrack$

$= \lbrack\cos{}72^{o} - \cos 36^{o}\rbrack$ $= \left\lbrack \frac{\sqrt{5} - 1}{4} - \frac{\sqrt{5} + 1}{4} \right\rbrack = \frac{- 1}{2}$.