Maclaurin Expansion: Obtaining First 3 Terms

[MisterMan]

Hi, I recently sat my Maths examination and there was a Maclaurin expansion question that I made an attempt at but I think it was wrong, it would be good if I could get help on this, it's too late to be of any real help but it will help me understand where I went wrong:

Obtain the first three non-zero terms in the Maclaurin expansion of (1 + sin²x)

What I done here was, let f(x) = (1 + sin²x) and differentiated until I got three non-zero terms when plugging in x = 0. But I'm not sure if this is "expandable" or whether I need to change sin²x into something that can be expanded.

Did I do the right thing or did I make a mistake?

 

[Dickfore]

The first term is 1.

 

[Hurkyl]

What you describe sounds like one of several correct ways to attack the problem. Of course, I cannot tell if you executed the attack correctly.

 

[Mark44]

A quicker way (and one probably alluded to by Hurkyl) is to take a couple of terms of the Maclaurin series for sin(x), square them, and then add 1.

 

[Dickfore]

Or use the trig formula

$$\sin^{2} x = \frac{1 - \cos{2 x}}{2}$$

and use the Maclaurin series for the cosine.