Laplace Transforms · Superchallenge problem

The Bessel function $J_0$ has Laplace transform $\mathcal{L}\{J_0(t)\} = \dfrac{1}{\sqrt{s^2+1}}$. Evaluate $\displaystyle\int_0^{\pi/2} J_0(\tau)\,J_0\!\left(\tfrac{\pi}{2} - \tau\right)d\tau$.