dumbish question: if cosine is simply sine but offset by pi/2 radians, why are we approximation these functions with an infinite sum of sines and cosines and not doing it just with sines?
If cosine is just sine rotated by 90°, then we really just use sines. Except we write cosine instead of a sine with a 90° rotation because it looks less confusing :)
The real goal is to arrive at the exponential representation, where the sines and cosines are absorbed into the exponential functions, which is later needed to write down the fourier integrals.
Your question boils down to preference. The two things you name are, in fact, equal. Math people just choose to write cos(x) instead of sin(x+pi/2)
Due to Euler's formula, there is a correspondence between a linear combination of e^(2 pi i n x) and e^(-2 pi i n x) and a linear combination of cos(2 pi n x) and sin(2 pi n x). In other words, both pairs of functions span the same subspace (with complex coefficients).
We need all of the exponentials to represent a periodic function, with positive and negative n, and so we need both cos and sin. If you leave out cos, you can only represent odd functions, and leaving out sin, even functions.
But: if your question is just why not just use sin(x) and sin(x-pi/2) instead of sin(x) and cos(x), well, it's the same thing. Just because you write cos(x) like sin(x-pi/2) doesn't mean it's not still cos(x)!
(Fun fact: cosine means "complementary sine," the sine of the complementary angle in the triangle.)
import numpy as np
a = np.pi/2
print(a)
> 1.5707963267948966
print(a == a + 1e-15)
> False
print(a == a + 1e-16)
> True
print(np.sin(1e-16))
> 1e-16
print(np.cos(np.pi/2+1e-16))
> 6.12323399574e-17
print(np.sin(1e-16) == np.cos(np.pi/2+1e-16))
> FalseBut there is already a different thing called a discrete cosine transform :)