Express the product of $$\cos 3x\cos 5x$$ as a sum or difference

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Uniteasy Admin · Accepted Answer

Apply Product to Sum identity for consine functions
$$\cos \alpha\cos \beta = \dfrac{1}{2} [\cos(\alpha+\beta ) +\cos(\alpha - \beta) ]$$
Here $$\alpha = 3x, \beta =5x $$
```katex
\begin{aligned} 
 \cos 3x\cos 5x &= \dfrac{1}{2} [\cos(3x+5x ) +\cos(3x - 5x) ] \ 
&=\dfrac{1}{2} [\cos 8x+\cos(-2x)]
 \end{aligned}
```
Apply the symmetry identity for cosine function
$$\cos(-2x) = \cos(2x)$$
Therefore,
$$\cos 3x\cos 5x = \dfrac{1}{2} (\cos 8x+\cos 2x)$$

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