  The origin of X-ray emission from the resolved kiloparsec-scale jets and
hotspots of many active galactic nuclei (AGN) remains uncertain, particularly
where the X-ray emission is separate from the radio-optical synchrotron
component. Possible explanations include synchrotron emission from a second
electron population and external Compton or synchrotron self-Compton processes
-- alternatives which imply very different physical conditions within the jet.
Until recently, X-ray studies of resolved jets and hotspots have been
restricted to below ~10 keV, often showing a hard spectral index indicating a
spectral peak beyond this energy range. Here we present NuSTAR observations of
the nearby powerful radio galaxy Pictor A, in which we clearly detect the
western hotspot at approximately 4' from the host galaxy, the most significant
detection of hotspot emission above 10 keV to date. The NuSTAR spectrum is best
fit by a single powerlaw of index $\Gamma$ = 2.03 $\pm$ 0.04; an exponential
cut-off gives a 1$\sigma$ lower limit on the cutoff energy of 40.7 keV. We
confirm previous findings of variations in the soft X-ray flux detected by
Chandra over the 2000 to 2015 period, at a significance of 6.5$\sigma$. This
rises to >8$\sigma$ in the common 3-8 keV band using the combined 22-year span
of Chandra and NuSTAR observations. The variability of the western Pictor A
hotspot strongly confirms the previously argued synchrotron nature of the X-ray
emission for the hotspot, while the lower bound to the spectral cutoff energy
implies electron energies in the hotspot reach up to at least a few TeV.
