Express the following as the product of sines and cosines: \[ \sin 5x – \sin x \]
Solution
Using the identity:
\[
\sin A – \sin B
=
2 \cos \frac{A+B}{2}
\sin \frac{A-B}{2}
\]
Here,
\[
A = 5x,\qquad B = x
\]
Substituting the values:
\[
\sin 5x – \sin x
=
2 \cos \frac{5x+x}{2}
\sin \frac{5x-x}{2}
\]
\[
=
2 \cos \frac{6x}{2}
\sin \frac{4x}{2}
\]
\[
=
2 \cos 3x \sin 2x
\]
Hence,
\[
\boxed{
\sin 5x – \sin x
=
2 \cos 3x \sin 2x
}
\]