Calculus/!!!!

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Question
In a model answer to a question I was trying to answer, there was an integrand with the following form: 0.5(1-cos(2x)) * 0.25[(1+cos(2x))^2]
then in the next step, i found the integrand changed to the following form:
         0.125[(1-((cos(2x))^2)) * (1+cos(2x))] !!!!
I don't know neither how nor why the integrand's form changed that way !!
I would really appreciate some assistance .
Thank You in advance.

Answer
Basically we are using the identity (a-b)*(a+b)=a^2 -b^2

Let's see how this works in your question:

0.5(1-cos(2x)) * 0.25[(1+cos(2x))^2] = 0.125* (1-cos(2x))*[(1+cos(2x))^2]

         = 0.125* (1-cos(2x))*(1+cos(2x)) *(1+cos(2x))

Looking at the above underlined part, if we refer to the identity first discussed,
we can say a=1  and  b =cos(2x)

So the underline part

(1-cos(2x))*(1+cos(2x)) becomes 1^2 - (cos(2x))^2 =1-(cos(2x))^2

Therefore, the overall expression is simply

0.125[(1-(cos(2x))^2) * (1+cos(2x))] (shown)

Hope this helps. Peace.  

Calculus

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