Solution
= Solution
Write $\cos x=1+u$ with $u=-x^2/2+x^4/24-x^6/720+O(x^8)$. Substitution into $\ln(1+u)=u-u^2/2+u^3/3+O(u^4)$ gives
$$
\ln\cos x=-\frac{x^2}{2}+\left(\frac1{24}-\frac18\right)x^4+\left(-\frac1{720}+\frac1{48}-\frac1{24}\right)x^6+O(x^8).
$$
Thus the first three nonzero terms of this even <Taylor series> are
$$
\boxed{\ln\cos x=-\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{45}+O(x^8).}
$$
The real logarithm is well defined near zero because $\cos x>0$ there.