I would like to expand functions of the form sin(ax), where a is an integer, to an expression only involving trigonometric functions with input x (such as sin(x), cos(x), sin(x)*cos(x), etc.).
I am aware I could do this by using each function's respective sum of angle formula, but that is a very tedious approach that can be very troublesome as "a" grows larger.
I am looking for some way to do it similar to the way (1+x)^n can be expanded by using Pascal's Triangle.
$$\cos(nx) = T_n(\cos(x))$$ where $T_n$ are the Chebyshev polynomials of the first kind. $$\sin(nx) = U_{n-1}(\cos(x)) \sin(x)$$ where $U_n$ are the Chebyshev polynomials of the second kind.