Firstly, I want to thank you for responding so appropriately.
Secondly this is a problem of my own creation. I do, however, have reason to believe it is not impossible.
You see this problem came from my own exploration. I have been observing the effects of functions in the form of T(x,y)=(x,y). That is to say having two inputs and two outputs. My goal is to have some function f(x) and some funxtion T(x,y) and somehow find p, the result of appling T(x,y) to every point defined by f(x). I call it p and not p(x) because the curve is usually not defined by a function.
The function T(x,y)=(x-y,x+y) results in a 45 degree rotation CCW, and keeps the origin in place (if applied to all points on the plane). The function f(x)=sin(x) is well ... you know. We know y=f(x) so T(x,y)=(x-f(x),x+f(x)) , expanding f(x) will result in T(x,y)=(x-sin(x),x+sin(x)). This is how I got the expressions earlier. By reason alone, it can be determined that the graph of the function which relates the expressions is a sine wave rotated CCW by 45 degrees. Thanks to this video https://youtu.be/BPgq2AudoEo that can be found easily. Such a curve can be defined as
y=(x*cos(a)+ y*sin(a))*sin(a)+sin(x*cos(a)+y*sin(a))*cos(a)
where a is the angle to rotate by, in this case 45 degrees, or parametricly
x=t*cos(a)-sin(t)*sin(a)
y=t*sin(a)+sin(t)*cos(a)
Although it is not a true function, because neither variable can be made independent, it defines the expected curve. This, at least I think, means that from the original post, there should be a way to reach the expected curve. I have no clue how you would reach the expected curve, I've tried, but I think there should be some way.
Reading this makes me feel as if I've posted this to the wrong category, but the original post fell into this category so I think it's fine.