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 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?