Cuprate high-T_c superconductors on the Mott-insulating side of optimal doping (with respect to the highest T_cs) exhibit enigmatic behavior in the non-superconducting state. Near optimal doping the transport and spectroscopic properties are unlike those of a Landau-Fermi liquid. For carrier concentrations below optimal doping a pseudogap removes quasi-particle spectral weight from parts of the Fermi surface, and causes a break-up of the Fermi surface into disconnected nodal and anti-nodal sectors. Here we show that the near-nodal excitations of underdoped cuprates obey Fermi liquid behavior. Our optical measurements reveal that the dynamical relaxation rate 1/tau(omega,T) collapses on a universal function proportional to (hbar omega)^2+(1.5 pi k_B T)^2. Hints at possible Fermi liquid behavior came from the recent discovery of quantum oscillations at low temperature and high magnetic field in underdoped YBa2Cu3O6+d and YBa2Cu4O8, from the observed T^2-dependence of the DC ({omega}=0) resistivity for both overdoped and underdoped cuprates, and from the two-fluid analysis of nuclear magnetic resonance data. However, the direct spectroscopic determination of the energy dependence of the life-time of the excitations -provided by our measurements- has been elusive up to now. This observation defies the standard lore of non-Fermi liquid physics in high T_c cuprates on the underdoped side of the phase diagram.
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