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In the following slide it shows how the taylor series of $(\cos x)^2$ is computed:

enter image description here

On the first line they simply take the taylor series of cosx and write it out twice, which makes sense. However, the order in which they multiply out the 1st line does not make sense to me.

For example, how did they get the 3rd term on the 2nd line: $(x^2/2!)^2$ ?

When I multiply out the 1st line I'm doing a straightforward distribution and I'm getting terms like this:

$$1 - (x^2/2!) + (x^4/4!) - (x^6/6!) - (x^2/2!) + (x^6/2!4!) + (x^{10}/2!6!) + (x^4/4!) - (x^6/4!2!) + (x^8/4!4!) - (x^{10}/4!6!)$$

When I combine the like terms above I do not end up with the 3rd line below and I was wondering if someone could point me in the right direction.

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    $\begingroup$ Collect terms in the "infinitive distributive product" according to increasing powers of $x$: First all terms of the form $\cdot x^0$, then all terms of the form $\cdot x^2$, and so on. Then you will get the third line (but not yet a general principle for the resulting coefficients). $\endgroup$ Commented May 27, 2013 at 18:02

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The terms are arranged as follows: $$ \begin{align} \cos^2 x &= (a_1 + a_2 + a_3 + a_4\ldots)(a_1 + a_2 + a_3 + a_4\ldots)\\ &= a_1^2 + 2a_1a_2 + a_2^2 + 2a_1a_3 + 2a_2a_3 + 2a_1a_4 + \ldots \end{align} $$

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