Facing problem in analysing Taylor series expansion

[bibhatsucool]

This is a very basic question .
Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it)
But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x.
Or if asked some other function such as Sin(f(x)) ,we just replace x by f(x) in Taylor expansion of Sin(x) ...
Instead of finding the expansion of g(x) =Sin(f(x)) term by term mechanically ?why ?
also I found it come out to be different... !
Help guys

 

[Orodruin]

It should not come out differently, if it does you are not doing it correctly. Note that the terms may look different in different sums, but essentially you should always be able to put it on the same form.