Contract the equatorial loop in $S^2$

Define paths \(f,g:I\to S^2\) by \[ f(s)=(\cos(2\pi s),\sin(2\pi s),0), \qquad g(s)=(1,0,0). \] Prove that \(f\) is homotopic to \(g\) relative to \(\{0,1\}\); that is, prove that the two loops are path-homotopic.