The value of \( \sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ \) is
Options:
(a) \( \frac12 \)
(b) \( -\frac12 \)
(c) \( -1 \)
(d) none of these
Solution:
\[
=\sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ
\]
Grouping terms,
\[
=(\sin78^\circ-\sin66^\circ)-(\sin42^\circ-\sin6^\circ)
\]
Using identity,
\[
\sin A-\sin B
=
2\cos\frac{A+B}{2}\sin\frac{A-B}{2}
\]
\[
=
2\cos72^\circ\sin6^\circ
–
2\cos24^\circ\sin18^\circ
\]
Using,
\[
2\sin A\cos B
=
\sin(A+B)+\sin(A-B)
\]
\[
=
(\sin78^\circ-\sin66^\circ)
–
(\sin42^\circ-\sin6^\circ)
\]
\[
=0
\]
Since \(0\) is not given in the options,
\[
\boxed{\text{none of these}}
\]
Correct option: (d)