Closed-form of an integral involving a Jacobi theta function, $ \int_0^{t} \theta_3(e^{-\pi^2 (t-\tau)}) \, \theta_2(e^{-\pi^2 \tau}) \ d\tau =1$

Question

Numerical calculation of a Duhamel-Integral coming up considering a unsteady state diffusion in a thin film electrode with zero initial concentration leads to the following strange identity:

$$ \int_0^t \theta_3(e^{-\pi^2 (t-\tau)}) \, \theta_2(e^{-\pi^2 \tau}) \ d \tau = 1$$

Question: Has anybody an idea to prove that.