Is there a way to skip every second term in the power series representation of $\sinh{x}$ and $\cosh{x}$ and adjust the other terms accordingly (approx.)?

So, instead of

$$\sinh{x} \approx x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \frac{x^9}{9!} + ... + \frac{x^{2n+1}}{(2n+1)!}$$ the following $$\sinh{x} \approx a_1 x + a_5\frac{x^5}{5!} + a_9 \frac{x^9}{9!} + ... + a_{4n+1}\frac{x^{4n+1}}{(4n+1)!}$$ and accordingly, instead of $$\cosh{x} \approx 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \frac{x^8}{8!} + ... + \frac{x^{2n}}{(2n)!}$$ the following $$\cosh{x} \approx a_0 + a_4\frac{x^4}{4!} + a_8 \frac{x^8}{8!} + ... + a_{4n}\frac{x^{4n}}{(4n)!}$$

The approximations do not need to be steady at $x = 0$