$$\int\cos(7x)\cos(17x)\cos(27x)\mathrm dx$$ I tried using the multiple angle identities but the working out was too tedious and hairy.

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    $\begingroup$ Hint: Use the product-to-sum formulae. $\endgroup$ – user137731 Jan 3 '16 at 21:10
  • $\begingroup$ If you're comfortable with the fact that $\cos(x)=\frac{e^{ix}+e^{-ix}}2$, this substitution is often helpful, since you can expand the products and end up with a sum of exponential functions, which are easy to integrate. (This is equivalent to using product-to-sum identities, but not so tricky to remember) $\endgroup$ – Milo Brandt Jan 3 '16 at 21:38
  • $\begingroup$ could you elaborate? @MiloBrandt $\endgroup$ – anja.wlotrzewiszczykowycki Jan 4 '16 at 7:13

We have:

$$\begin{cases} \cos(a+b) = \cos a \cos b - \sin a \sin b \\ \cos(a-b) = \cos a \cos b + \sin a \sin b \end{cases}$$

Adding the two equations:

$$\cos a \cos b = \frac12 \left( \cos(a+b) + \cos(a-b) \right)$$

You will need to apply this formula three times.


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