Limited developement of log and trigonomic expression

Question

What is the best way to calculate the limited developement of

$f(x) = \log( 2 \cos(x) + \sin(x) )$ In the neighborhood of $0$ at the order $4$

Any ideas?

Answer 1

Hint

write $f(x)$ as

$$\ln(2)+\ln\left(1-2(\sin( \frac{x}{2} ))^2+\frac{\sin(x)}{2}\right)$$.

and use

$$\sin(t)=t-\frac{t^3}{6}+t^4\epsilon_1(t)$$

$$\ln(1-X)=-X+\frac{X^2}{2}-\frac{X^3}{3}+\frac{X^4}{4}+X^4\epsilon_2(X)$$