Result
R2484: Moments of a Bernoulli random boolean number shows that $\mathbb{E} X = \mathbb{E} X^2 = \theta$. Using
R2262: Real variance partition into first and second moments, we then have
\begin{equation}
\mathsf{Var} X
= \mathbb{E} X^2 - (\mathbb{E} X)^2
= \theta - \theta^2
= \theta (1 - \theta)
\end{equation}
$\square$
