In this paper, we derive an integral representation for the density of the reciprocal of the first hitting time of the boundary of a wedge of angle $\pi/4$ by a radial Dunkl process with equal multiplicity values. Not only this representation readily yields the non negativity of the density, but also provides an analogue of Dufresne's result on the distribution of the first hitting time of zero by a Bessel process and a generalization of the Vakeroudis-Yor's identity satisfied by the first exit time from a wedge by a planar Brownian motion. We also use a result due to Spitzer on the angular part of the planar Brownian motion to prove a representation of the tail distribution of its first exit time from a dihedral wedge through the square wave function.