A special case of softmax regression with K=2: (due to the overparameterization property)

$$\begin{align} h_\theta(x) &= \frac{1}{ \exp(\theta^{(1)\top}x) + \exp( \theta^{(2)\top} x^{(i)} ) } \begin{bmatrix} \exp( \theta^{(1)\top} x ) \\ \exp( \theta^{(2)\top} x ) \end{bmatrix}\\ &= \frac{1}{ \exp( (\theta^{(1)}-\theta^{(2)})^\top x^{(i)} ) + \exp(\vec{0}^\top x) } \begin{bmatrix} \exp( (\theta^{(1)}-\theta^{(2)})^\top x ) \exp( \vec{0}^\top x ) \\ \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{ 1 + \exp( (\theta^{(1)}-\theta^{(2)})^\top x^{(i)} ) } \\ \frac{\exp( (\theta^{(1)}-\theta^{(2)})^\top x )}{ 1 + \exp( (\theta^{(1)}-\theta^{(2)})^\top x^{(i)} ) } \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{ 1 + \exp( (\theta^{(1)}-\theta^{(2)})^\top x^{(i)} ) } \\ 1 - \frac{1}{ 1 + \exp( (\theta^{(1)}-\theta^{(2)})^\top x^{(i)} ) } \\ \end{bmatrix} \end{align}$$
Finally:
$$\begin{align} P(y=1|x) &= h_\theta(x) = \frac{1}{1 + \exp(-\theta^\top x)}\\ P(y=0|x) &= 1 - P(y=1|x) = 1 - h_\theta(x). \end{align}$$
Sigmoid function... (left to be expanded)
$$f(z) = \frac{1}{1+\exp(-z)}.$$

Reference

http://ufldl.stanford.edu/tutorial/supervised/LogisticRegression/
http://ufldl.stanford.edu/tutorial/supervised/SoftmaxRegression/