A simple path homotopy

The two half-circles can be parametrized as follows:
$$\gamma_{0}(s)=(\cos(s),\sin(s)),\;\gamma_{1}(s)=(\cos(-s),\sin(-s))=(\cos(s),-\sin(s))$$(note that we use the letter $s$ because the letter $t$ will be used for the homotopy parameter).

So a natural path homotopy is:
$$H(t,s)=\left(\cos(s),(1-2t)\sin(s)\right).$$