The existing proofs of the Kuznetsov trace formula appeal to subtle properties of special functions in non-standard ranges. As a consequence, one often has to confront cumbersome oscillatory integrals in concrete applications. In this paper, the authors state and derive the Kuznetsov trace formula from the pretrace formula, that is, from the spectral decomposition of the automorphic kernel \(K(z,w)\). The approach here requires no prior knowledge of special functions except real analysis and the definitions \( K_{it}(x) = \int_0^\infty e^{-x \cosh(v)} \cos(tv) \;\mathrm{d}v\) and \(J_0(x) = \frac{1}{2\pi} \int_0^{2\pi} \cos(x \cos(\theta)) \;\mathrm{d}\theta\). In this new formulation, the original oscillatory integrals reduce to something close to the composition of two Fourier transforms. In Section 4, the equivalence between the new and the classical formulation is established. In Section 5, the authors give estimates and explicit instances of the Kuznetsov formula to illustrate some advantages of their new formulation.