Find the Maclaurin series for the function $f(x) = cos(x) sin(x)$ up to the fourth order term.

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the Maclaurin series for $f(x) = \cos(x) \sin(x)$ up to the fourth order term, we start by expanding both $\cos(x)$ and $\sin(x)$ using their Maclaurin series. ‖ ‖ step 2 ⋮ The Maclaurin series for $\cos(x)$ is: $$ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6) $$ ‖ ‖ step 3 ⋮ The Maclaurin series for $\sin(x)$ is: $$ \sin(x) = x - \frac{x^3}{3!} + O(x^5) $$ ‖ ‖ step 4 ⋮ Multiply the series for $\cos(x)$ and $\sin(x)$ and collect terms up to $x^4$: $$ \cos(x) \sin(x) = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!}\right) \left(x - \frac{x^3}{3!}\right) $$ ‖ ‖ step 5 ⋮ Expanding and simplifying the product: $$ \cos(x) \sin(x) = x - \frac{x^3}{2} - \frac{x^3}{6} + \frac{x^5}{24} + O(x^5) $$ $$ = x - \frac{2x^3}{3} + O(x^5) $$ ‖ ∻Answer∻ ⚹ The Maclaurin series for $f(x) = \cos(x) \sin(x)$ up to the fourth order term is: $$ f(x) = x - \frac{2x^3}{3} + O(x^5) $$ ⚹ ∻Key Concept∻ ⚹Maclaurin series expansion⚹ ∻Explanation∻ ⚹The Maclaurin series expansion is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point (usually zero). In this case, we expanded $\cos(x)$ and $\sin(x)$ and multiplied the series to find the series for $\cos(x) \sin(x)$.⚹◊What is the formula for the nth term of the Maclaurin series for $cos(x)$ and $sin(x)$?⍭ Generate me a similar question◊

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