\( \cos52^\circ+\cos68^\circ+\cos172^\circ \)
Options:
(a) \(0\)
(b) \(1\)
(c) \(2\)
(d) \( \frac32 \)
Solution:
Using,
\[
\cos(180^\circ-\theta)=-\cos\theta
\]
\[
\cos172^\circ=-\cos8^\circ
\]
Therefore,
\[
\cos52^\circ+\cos68^\circ+\cos172^\circ
\]
\[
=\cos52^\circ+\cos68^\circ-\cos8^\circ
\]
Using identity,
\[
\cos A+\cos B
=
2\cos\frac{A+B}{2}\cos\frac{A-B}{2}
\]
\[
=
2\cos60^\circ\cos8^\circ-\cos8^\circ
\]
\[
=
2\left(\frac12\right)\cos8^\circ-\cos8^\circ
\]
\[
=
\cos8^\circ-\cos8^\circ
\]
\[
=0
\]
\[
\boxed{0}
\]
Correct option: (a)