existence of a contraction of sin(x) on $[-\frac{\pi}{2},\frac{\pi}{2}]$

Question

I tried to proof whether a contraction with sin(x) on $D=[-\frac{\pi}{2},\frac{\pi}{2}]$ can happen. It seems that there exists no contraction because the contraction constant L leads to $L=max\rvert f'\lvert=1$. Are you guys agree with this result?

Best regards, gui