\( \cos35^\circ+\cos85^\circ+\cos155^\circ \)
Options:
(a) \(0\)
(b) \( \frac1{\sqrt3} \)
(c) \( \frac1{\sqrt2} \)
(d) \( \cos275^\circ \)
Solution:
\[
=\cos35^\circ+\cos85^\circ+\cos155^\circ
\]
Using,
\[
\cos(180^\circ-\theta)=-\cos\theta
\]
\[
=\cos35^\circ+\cos85^\circ-\cos25^\circ
\]
Using identity,
\[
\cos A+\cos B
=
2\cos\frac{A+B}{2}\cos\frac{A-B}{2}
\]
\[
=
2\cos60^\circ\cos25^\circ-\cos25^\circ
\]
\[
=
2\left(\frac12\right)\cos25^\circ-\cos25^\circ
\]
\[
=
\cos25^\circ-\cos25^\circ
\]
\[
=0
\]
\[
\boxed{0}
\]
Correct option: (a)