I am faced with the following problem:
Let $f = (1 + a)^3\sin x \cos x - e^{3x}\cos^2 (2x)$. Find a representation of $f$ in the form $f = a + bx + cx^2$ (a, b, c numerical constant) valid for $x \ll1$. You may assume that $e^x = 1 + x + \dfrac{x^2}{2}$, $\cos x = 1 - \dfrac{x^2}{2}$ and $\sin x = x$
I understand that to find the required form of $f$, I would need to formulate the Taylor series of $f$. Because we are told that $x \ll 1$, I would need to formulate the Taylor series around $x = 0$. If I understand this problem correctly, this is all that needs to be taken into account and we may solve the problem with this information alone. My question however is, of what use are the last three assumptions in solving the problem?
Is this a Taylor series problem? Yes and No. Yes, in the sense that there is content in the problem that is related to the Taylor series. No, in the sense that you do not need to calculate the Taylor series.
The assumptions are given so that you may substitute the assumptions into the overall function $f$ after a small amount of algebraic and trigonometric manipulation instead of calculating the Taylor series from first principles. This gives:
$f = (1 + a)^3(assumption3)(assumption2) - (assumption1)^{3}[1 - 2(assumption3)^2 x]^2$
$f = (1 + a)^3(x)(1 - \tfrac{x^2}{2}) - (1 + x + \tfrac{x^2}{2})^{3}[1 - 2(x)^2 x]^2$
You will notice that the assumptions are themselves a Taylor series of the second degree - you could work this out yourself, but its given for you.