Question Maclaurin series

Liveit

Right I don't understand one bit so I would appreciate if someone helped.
So part (i) of the question is
Derive the MacLaurin series for f(x)=cosx, up to and including the term containing x^4.
The answer to that is



(ii)Hence or otherwise, show that the first three non-zero terms of the Maclaurin series for f(x)= are
.

the answer starts as this and follows on to the answer above.


My question is how does he get the 4x^2 and 16x^2 because I thought that it would be 2x^2 and 2x^2 in their place since it is cos2x instead of cosx above, and i thought you would just multiply by two.. Anyone know?
Thanks

mathew

Instead of x being the argument in the cosine its now 2x (from the trig identity)
So you have to put 2x in the place of x in your MacLaurin series.
the second term has x^2, if you square 2x you get 4x^2
the third term has x^4, if you raise 2x to the 4th power you get 16x^4

Liveit

Ah that didn't occur to me at all, better it is a simple mistake than a complicated one i suppose.

Fremen

Man alive, why not just square the power series and keep terms up to fourth order?
(This is personal preference. I'd almost always take a general method over a specific one.)