\( \sin163^\circ\cos347^\circ+\sin73^\circ\sin167^\circ \)
Options:
(a) \(0\)
(b) \(\frac12\)
(c) \(1\)
(d) none of these
Solution:
\[
=\sin163^\circ\cos347^\circ+\sin73^\circ\sin167^\circ
\]
\[
=\sin17^\circ\cos13^\circ+\sin73^\circ\sin13^\circ
\]
\[
=\sin17^\circ\cos13^\circ+\cos17^\circ\sin13^\circ
\]
Using identity,
\[
\sin A\cos B+\cos A\sin B=\sin(A+B)
\]
\[
=\sin(17^\circ+13^\circ)
\]
\[
=\sin30^\circ
\]
\[
=\frac12
\]
\[
\boxed{\frac12}
\]
Correct option: (b)