Taylor Polynomial of $f(x)=\cos(x)\cdot\sin(x)$

Question

How would I calculate the third maclaurin/taylor polynomial on $\cos(a) \cdot \sin(b)$, Do I use the product rule when I calculate the derivatives? I don't know where to start or read about it, been stuck for a couple of days now. Thank you!

Answer 1

The easieast way was pointed out by Kim Jong Un in the comment, but there are a couple of other ways to do that. Remember that the $n$-th degree McLaurin polynomial is

$$T_n(x) = \sum_{i=0}^n f^{(n)}(0) \frac{x^n}{n!}$$

But if $f(x) = \sin (x) \cos (x)$ then $f'(x) = \cos^2 (x) - \sin^2(x)$, $f''(x) = -4\sin(x)\cos(x)$ and $f'''(x) = -4( \cos^2 (x) - \sin^2(x))$

Then, $f(0) = 0$, $f'(0) = 1$, $f''(0)=0$, $f'''(0)=-4$

So $$T_3(x) = x -\frac{2}{3}x^3$$