This is an annoying but routine algebra problem (as you probably have already guessed). Let's call the right-hand side of your first equation $L(\vec{x})$. Then we can cancel the factors of $1/\sqrt{2\pi\sigma^2}$ to get
$$L(\vec{x}) = {\exp \left( - {1 \over 2} \sum_{i=1}^n (x_i-1)^2 \right) \over \exp \left( - {1 \over 2} \sum_{i=1}^n x_i^2 \right)}$$
It is often a useful trick, in these likelihood-ratio problems, to rewrite a ratio of exponentials as an exponential of a difference; that gives
$$L(\vec{x}) = \exp \left( -{1 \over 2} \left[ \sum_{i=1}^n (x_i-1)^2 - \sum_{i=1}^n x_i^2 \right] \right)$$
and now you can combine the two sums into one to get
$$L(\vec{x}) = \exp \left( -{1 \over 2} \left[ \sum_{i=1}^n ((x_i-1)^2 - x_i^2) \right] \right)$$
The summand $(x_i-1)^2 - x_i^2$ is equal to $-2x_i + 1$ and so
$$L(\vec{x}) = \exp \left( -{1 \over 2} \left[ \sum_{i=1}^n (-2x_i + 1) \right] \right)$$
and you should be able to do it from here.