Surely, one can compose a power series for them, and any partial sum of those series would be defined, But are they defined in the limit?

I mean, what is $\cos \omega$, for instance?

Does the trigonometric equality $\cos^2 x+\sin^2 x=1$ still hold?

P.S. More context. [This Wikipedia article][1] on Hardy fields says "*This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields.*". Yet, surreal numbers are an H-field, that is a Hardy field with unity. The Wikipedia is wrong?


  [1]: https://en.wikipedia.org/wiki/Hardy_field