Question

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

Collected in the board: Trigonometry

Steven Zheng posted 2 months ago


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


\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)


Steven Zheng posted 2 months ago

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