Express the following as the product of sines and cosines: \[ \cos 12x + \cos 8x \]
Solution
Using the identity:
\[
\cos A + \cos B
=
2 \cos \frac{A+B}{2}
\cos \frac{A-B}{2}
\]
Here,
\[
A = 12x,\qquad B = 8x
\]
Substituting the values:
\[
\cos 12x + \cos 8x
=
2 \cos \frac{12x+8x}{2}
\cos \frac{12x-8x}{2}
\]
\[
=
2 \cos \frac{20x}{2}
\cos \frac{4x}{2}
\]
\[
=
2 \cos 10x \cos 2x
\]
Hence,
\[
\boxed{
\cos 12x + \cos 8x
=
2 \cos 10x \cos 2x
}
\]