The value of \( \sin\frac{\pi}{18}+\sin\frac{\pi}{9}+\sin\frac{2\pi}{9}+\sin\frac{5\pi}{18} \) is given by
Options:
(a) \( \sin\frac{7\pi}{18}+\sin\frac{4\pi}{9} \)
(b) \(1\)
(c) \( \cos\frac{\pi}{6}+\cos\frac{3\pi}{7} \)
(d) \( \cos\frac{\pi}{9}+\sin\frac{\pi}{9} \)
Solution:
\[
=\sin\frac{\pi}{18}+\sin\frac{5\pi}{18}
+
\sin\frac{\pi}{9}+\sin\frac{2\pi}{9}
\]
Using identity,
\[
\sin A+\sin B
=
2\sin\frac{A+B}{2}\cos\frac{A-B}{2}
\]
\[
=
2\sin\frac{6\pi}{36}\cos\frac{4\pi}{36}
+
2\sin\frac{3\pi}{18}\cos\frac{\pi}{18}
\]
\[
=
2\sin\frac{\pi}{6}\cos\frac{\pi}{9}
+
2\sin\frac{\pi}{6}\cos\frac{\pi}{18}
\]
Since,
\[
2\sin\frac{\pi}{6}=1
\]
\[
=
\cos\frac{\pi}{9}+\cos\frac{\pi}{18}
\]
Using,
\[
\cos\frac{\pi}{18}=\sin\left(\frac{\pi}{2}-\frac{\pi}{18}\right)
=\sin\frac{4\pi}{9}
\]
\[
=
\cos\frac{\pi}{9}+\sin\frac{4\pi}{9}
\]
\[
=
\sin\frac{7\pi}{18}+\sin\frac{4\pi}{9}
\]
\[
\boxed{\sin\frac{7\pi}{18}+\sin\frac{4\pi}{9}}
\]
Correct option: (a)