There are two methods.

1) Let #f(x)=cos^2(x)# and use #f(0)+f'(0)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+\cdots#

We have, by the Chain Rule and/or Product Rule,

#f'(x)=-2cos(x)sin(x)#, #f''(x)=2sin^2(x)-2cos^2(x)#

#f'''(x)=4sin(x)cos(x)+4cos(x)sin(x)=8cos(x)sin(x)#

#f''''(x)=-8sin^2(x)+8cos^2(x)#, #f'''''(x)=\cdots=-32cos(x)sin(x)#,

#f''''''(x)=32sin^2(x)-32cos^2(x)#, etc...

Hence, #f(0)=1#, #f'(0)=0#, #f''(0)=-2#, #f'''(0)=0#, #f''''(0)=8#, #f'''''(0)=0#, #f''''''(0)=-32#, etc...

Since #2! =2#, #8/(4!)=8/24=1/3#, and #(-32)/(6!)=(-32)/720=-2/45#, this much calculation leads to an answer of

#1-x^2+x^4/3-2/45 x^6+\cdots#

This does happen to converge for all #x# and it does happen to equal #cos^2(x)# for all #x#. You can also check on your own that the next non-zero term is #+x^8/315#

2) Use the well-known Maclaurin series #cos(x)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+\cdots#

#=1-x^2/2+x^4/24-x^6/720+\cdots# and multiply it by itself (square it).

To do this, first multiply the first term #1# by everything in the series to get

#cos^2(x)=(1-x^2/2+x^4/24-x^6/720+\cdots)+\cdots#

Next, multiply #-x^2/2# by everything in the series to get

#cos^2(x)=(1-x^2/2+x^4/24-x^6/720+\cdots)+(-x^2/2+x^4/4-x^6/48+x^8/1440+\cdots)+\cdots#

Then multiply #x^4/24# by everything in the series to get

#cos^2(x)=(1-x^2/2+x^4/24-x^6/720+\cdots)+(-x^2/2+x^4/4-x^6/48+x^8/1440+\cdots)+(x^4/24-x^6/48+x^8/576-x^10/17280+\cdots)+\cdots#

etc...

If you go out far enough and combine "like-terms", using the facts, for instance, that #-1/2-1/2=-1# and #1/24+1/4+1/24=8/24=1/3#, etc..., you'll eventually come to the same answer as above:

#1-x^2+x^4/3-2/45 x^6+\cdots#

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