Equivalent infinitesimal for $\log(\cos(x))$

Question

I recently came across a limit problem which replaces $\log(\cos(x))$ with an equivalent infinitesimal $\cos(x) - 1$.

How do we prove that $\cos(x)-1$ is the equivalent infinitesimal for $\log(\cos(x))$?

Answer 1

Write

$$ \ln(\cos(x))=\ln(1+\cos(x)-1)=\cos(x)-1 +o(\cos(x)-1)$$