Express the following as the product of sines and cosines: \[ \cos 12x – \cos 4x \]
Solution
Using the identity:
\[
\cos A – \cos B
=
-2 \sin \frac{A+B}{2}
\sin \frac{A-B}{2}
\]
Here,
\[
A = 12x,\qquad B = 4x
\]
Substituting the values:
\[
\cos 12x – \cos 4x
=
-2 \sin \frac{12x+4x}{2}
\sin \frac{12x-4x}{2}
\]
\[
=
-2 \sin \frac{16x}{2}
\sin \frac{8x}{2}
\]
\[
=
-2 \sin 8x \sin 4x
\]
Hence,
\[
\boxed{
\cos 12x – \cos 4x
=
-2 \sin 8x \sin 4x
}
\]