A rational approximation of cosine

Expansion of the fraction.
$$\frac{1+ax^2}{1+bx^2}=1+(a-b)x^2-b(a-b)x^4+o(x^4).$$

Matching with $\cos x=1-\frac{x^2}2+\frac{x^4}{24}+o(x^4)$.
$$a-b=-\frac12,\qquad -b(a-b)=\frac b2=\frac1{24}\ \Longrightarrow\ b=\frac1{12},\ a=-\frac5{12}.$$

Result. $\cos x\approx\dfrac{1-\frac5{12}x^2}{1+\frac1{12}x^2}=\dfrac{12-5x^2}{12+x^2}$. Pushing the computation to order $6$ gives
$$\cos x-\frac{12-5x^2}{12+x^2}=\frac{x^6}{480}+o(x^6).$$

Test. At $x=1$: $\frac7{13}=0.5385$ while $\cos1=0.5403$, an error of $1.8\cdot10^{-3}$, close to $\frac1{480}\approx2\cdot10^{-3}$. The Taylor polynomial of degree $4$ gives $0.54167$, with a similar error.

Remark. With the same number of coefficients (three), the rational approximation has the same order of accuracy as the polynomial. Padé approximants shine for functions with poles, where polynomials do badly.