Computing taylor series for trigonometric exponential function How do I compute the taylor series for $\cos(x)^{\sin(x)}$ ? I tried using the $e^x$ rule but I still am not getting to the result:
$$\cos(x)^{\sin(x)}=1-\frac{x^3}{2}+\frac{x^6}{8}+o(x^6).$$

Your formula ($\cos(x)^{\sin(x)}=1-\frac{x^3}{2}+\frac{x^6}{8}+o(x^6).$) has been achieved from the definition of The Taylor Series:
$$f(x) = \sum_{i=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$
Where $f^{(n)}(x)$ is $n$th derivative of $f(x)$ with respect to $x$.
(Notice that $f\in c^{\infty}$)
put $x_0=0$ and calculate the coefficients of $x^0$, $x^1$, ... $x^6$.
