Oh jeez I'm being a dumb ass. I forgot your original question which was whether you could express sin, cos and tan in terms of basic arithmetic.

I showed you how to do it with sin and cos.

Then I got lost in the Maclaurin series. This is useful:

tan(x) = sin(x) / cos(x)

That is, the tangent of an angle is equal to its sine divided by its cosine. Since we can express the sine and cosine as Maclaurin series, we can express the tangent as one Maclaurin series divided by another:

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

cos(x) = 1 - x^2/2! + x^4/4! -x^6/6! + x^8/8! - ...

and tan(x) = sin(x) / cos(x)

so tan(x) = (x - x^3/3! + x^5/5! - x^7/7! + ...) / (1 - x^2/2! + x^4/4! -x^6/6! + x^8/8! - ...)

I hope that makes sense