Estimate the error in replacing $\cos{x}/\cos{y}$ by $\frac{1}{2} > \left(y^2-x^2\right)+1$ for $|x|,|y|<\pi/6$.
Answer takes the fourth term in the Taylor expansion $\frac{1}{24} \left(x^4-6 x^2 y^2+5 y^4\right)$ and says its max value for $|x|,|y|<\pi/6$ is the error.
I'm a bit confused by this, as I would have thought a proper estimation uses the remainder term, which is of order $4$, which is not the same as just the fourth term. Can anyone explain why this is justified?