Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[fx=\sin{x},\sqrt{3}fx-f(\frac{\pi }{2}+{x}^{2})\]
Break down the problem into these 2 equations.
\[fx=\sin{x}\]
\[fx=\sqrt{3}fx-f(\frac{\pi }{2}+{x}^{2})\]
Solve the 1st equation: \(fx=\sin{x}\).
Divide both sides by \(x\).
\[f=\frac{\sin{x}}{x}\]
\[f=\frac{\sin{x}}{x}\]
Solve the 2nd equation: \(fx=\sqrt{3}fx-f(\frac{\pi }{2}+{x}^{2})\).
Expand.
\[fx=\sqrt{3}fx-\frac{f\pi }{2}-f{x}^{2}\]
Multiply both sides by \(2\).
\[2fx=2\sqrt{3}fx-f\pi -2f{x}^{2}\]
Move all terms to one side.
\[2fx-2\sqrt{3}fx+f\pi +2f{x}^{2}=0\]
Factor out the common term \(f\).
\[f(2x-2\sqrt{3}x+\pi +2{x}^{2})=0\]
Divide both sides by \(2x-2\sqrt{3}x+\pi +2{x}^{2}\).
\[f=0\]
\[f=0\]
Collect all solutions.
\[f=\frac{\sin{x}}{x},0\]
f=sin(x)/x,0
