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For a loop $a$ around $x$ and $\overline{a}$ its opposite, I want to construct an homotopy between $a \cdot \overline a$ and $c_x$. The usual solution is the one that can be found for example here.

However I was wondering if there is anything wrong with another one, namely $H(x,t)= \left\{\begin{array}{lr} a(2tx) & 0\leq x\leq 1/2\\ a(2t-2tx) & 1/2\leq x\leq 1\ \end{array} \right |?$

If no, then is there any benefit to using the usual one?

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    Yours is a fine homotopy. I don't see an advantage to the other one, but it's possible that it is more convenient for some things I don't currently think of. – 2017-02-12

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As Daniel Fischer noted, the given homotopy is also viable.