<s>[INST] Expansion of $\sin x$ I wanted to know, how can I derive:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$
 [/INST] One could use Euler's Theorem to get $$\cos x + i\sin x=e^{ix}=\sum_{k=0}^\infty \frac{(ix)^k}{k!}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots  \right)$$ Now equating real and imaginary parts you will get the series expansion for both $\cos x$ and $\sin x$.
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