Question #faa96

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Answer 1 · 2016-04-12T13:36:44

#2(cos(4alpha)+1)^2#

We should initially rewrite #cos(8alpha)# using the following cosine double-angle identity:

#cos(2x)=2cos^2(x)-1#

We can apply this to #cos(8alpha)# as follows:

#cos(8alpha)=2cos^2(4alpha)-1#

Plugging this into the original expression, we see that it equals

#3+4cos(4alpha)+2cos^2(4alpha)-1#

#=2+4cos(4alpha)+2cos^2(4alpha)#

Factor a #2# from each term and rearrange order.

#=2(cos^2(4alpha)+2cos(4alpha)+1)#

Finally note that #c^2+2c+1 = (c+1)^2#, so we can write

#=2(cos(4alpha)+1)^2#