The Value of \( \cos12^\circ+\cos84^\circ+\cos156^\circ+\cos132^\circ \)
Question
Find the value of
\[ \cos12^\circ+\cos84^\circ+\cos156^\circ+\cos132^\circ \]
(a) \(\frac12\)
(b) \(1\)
(c) \(-\frac12\)
(d) \(\frac18\)
Solution
Use the identity
\[ \cos(180^\circ-\theta)=-\cos\theta \]
Therefore,
\[ \cos156^\circ = -\cos24^\circ \]
and
\[ \cos132^\circ = -\cos48^\circ \]
Hence,
\[ \cos12^\circ+\cos84^\circ+\cos156^\circ+\cos132^\circ \]
\[ = \cos12^\circ+\cos84^\circ-\cos24^\circ-\cos48^\circ \]
Now,
\[ \cos12^\circ+\cos84^\circ = 2\cos48^\circ\cos36^\circ \]
and
\[ \cos24^\circ+\cos48^\circ = 2\cos36^\circ\cos12^\circ \]
Therefore,
\[ = 2\cos36^\circ(\cos48^\circ-\cos12^\circ) \]
Using
\[ \cos C-\cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \]
\[ = 2\cos36^\circ \left[-2\sin30^\circ\sin18^\circ\right] \]
\[ = -2\cos36^\circ\sin18^\circ \]
Using the exact values
\[ \cos36^\circ=\frac{\sqrt5+1}{4}, \qquad \sin18^\circ=\frac{\sqrt5-1}{4} \]
\[ = -2\cdot \frac{(\sqrt5+1)(\sqrt5-1)}{16} \]
\[ = -2\cdot\frac{4}{16} \]
\[ = -\frac12 \]
Final Answer
\[ \boxed{-\frac12} \]
Hence, the correct option is (c) \(-\frac12\).