Question

how to get the other maclaurin series from the basic ones such as e^x, sin(x), cos(x), 1/(1-x) , ln(1-x)

Answer 1

well for sin is a derivative of cosine so you need know the series of sin and take a derivative of it to find cosine. same with tan you can find from dividing series of sin by cosine. i think this page will help: http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx

Answer 2

There are lots of things you can do:
1. substitute values, for example,
e^(x^2) = 1+(x^2)+(x^2)^2/2!+(x^2)^3/3!+...
1/(1-sqrt(x)) = 1+sqrt(x)+x+xsqrt(x)+x^2+....
2. add or subtract
1+x+x^2+x^3+...+x^k = 1/(1-x) - x^(k+1)/(1-x)
cosh(x)=(1/2)(e^x+e^(-x))
3. multiply or divide
4. differentiate or integrate (term by term)