Q:

Trigonometry equality

Is there any chance to find $A$ and $T$ such that
$$a\sin(x) + b\cos(x) = A\cdot T(x)$$ where $T(x)$ is a product of trigonometric functions and neither $a$ or $b$ appear in $T$?

A:

We interpret your question to ask whether there is a function $A(s,t)$ and a function $T(x)$ such that 
$$s\sin x+t\cos x=A(s,t)T(x)$$
for all $s,t,x$. 
Take the partial derivative of both sides with respect to $s$. We get
$$\sin x=T(x)\frac{\partial}{\partial s}A(s,t)\tag{1}$$
identically in $x$. Similarly
$$\cos x=T(x)\frac{\partial}{\partial t}A(s,t).\tag{2}$$
This is impossible, since the left side of (1) is $0$ at $x=0$, while the left side of (1) is not. 

