Find all positive differentiable functions $f$ that satisfy$$\int_0^x \sin(t) f(t) dt = [f(x)]^2-1$$for all real numbers $x$.
I know this question has been answered here: Proof for differentiable functions, but I'm still a little confused about parts of the solution. I mostly followed along hamam_Abdallah's solution. I mainly do not understand why we need to first differentiate both sides to get $f(x)sin(x)=2f(x)f'(x)$ if we're trying to go backwards from integration.
Sorry in advance if this is very basic; I don't have much background in integration!