\( \sin47^\circ+\sin61^\circ-\sin11^\circ-\sin25^\circ \) is equal to
Options:
(a) \( \sin36^\circ \)
(b) \( \cos36^\circ \)
(c) \( \sin7^\circ \)
(d) \( \cos7^\circ \)
Solution:
\[
=(\sin47^\circ-\sin11^\circ)+(\sin61^\circ-\sin25^\circ)
\]
Using identity,
\[
\sin A-\sin B
=
2\cos\frac{A+B}{2}\sin\frac{A-B}{2}
\]
\[
=
2\cos29^\circ\sin18^\circ
+
2\cos43^\circ\sin18^\circ
\]
\[
=
2\sin18^\circ(\cos29^\circ+\cos43^\circ)
\]
Using identity,
\[
\cos A+\cos B
=
2\cos\frac{A+B}{2}\cos\frac{A-B}{2}
\]
\[
=
2\sin18^\circ
\left(
2\cos36^\circ\cos7^\circ
\right)
\]
\[
=
4\sin18^\circ\cos36^\circ\cos7^\circ
\]
Using,
\[
2\sin18^\circ\cos36^\circ=\sin54^\circ-\sin18^\circ
\]
\[
=
2\cos7^\circ
\left(
\sin54^\circ-\sin18^\circ
\right)
\]
Since,
\[
\sin54^\circ=\cos36^\circ
\]
and
\[
\sin18^\circ=\frac{\sqrt5-1}{4}
\]
we get
\[
= \cos7^\circ
\]
\[
\boxed{\cos7^\circ}
\]
Correct option: (d)