\begin{align} \sin x &=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}+o(x^{10}) \\ \cos x&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+o(x^9) \\ e^x &=1+ x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+o(x^5) \\ e^{ix} &= 1 + ix -\frac{x^2}{2!} – \frac{ix^3}{3!} +\frac{x^4}{4!}+\frac{ix^5}{5!} …\\ &=\cos x + i\sin x\\ (1-x)^{-1}=\frac{1}{1-x} &=1+x+x^2+x^3+x^4 … \\ (1+x)^{-1}=\frac{1}{1+x} &=1-x+x^2-x^3+x^4 … \\ \ln(1-x)&=-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4} …\\ \ln(1+x)&=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}…\\ \tan x &=x+\frac{x^3}{3}+\frac{2x^5}{15}+o(x^6) \\ \arctan x&=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\frac{1}{9}x^9+o(x^{10}) \\ \arctan^{‘}x&=1-x^2+x^4-x^6+x^8 … \\ arccot x&=-x+\frac{1}{3}x^3-\frac{1}{5}x^5+\frac{1}{7}x^7-\frac{1}{9}x^9+o(x^{10}) \\ \end{align}