The value of \( \sin50^\circ-\sin70^\circ+\sin10^\circ \) is equal to
Options:
(a) \(1\)
(b) \(0\)
(c) \( \frac12 \)
(d) \(2\)
Solution:
\[
=\sin50^\circ-\sin70^\circ+\sin10^\circ
\]
Grouping first two terms,
\[
=(\sin50^\circ-\sin70^\circ)+\sin10^\circ
\]
Using identity,
\[
\sin A-\sin B
=
2\cos\frac{A+B}{2}\sin\frac{A-B}{2}
\]
\[
=
2\cos60^\circ\sin(-10^\circ)+\sin10^\circ
\]
\[
=
2\left(\frac12\right)(-\sin10^\circ)+\sin10^\circ
\]
\[
=
-\sin10^\circ+\sin10^\circ
\]
\[
=0
\]
\[
\boxed{0}
\]
Correct option: (b)