  We present simulated J-band spectroscopy of red giants and supergiants with a
42m European Extremely Large Telescope (E-ELT), using tools developed toward
the EAGLE Phase A instrument study. The simulated spectra are used to
demonstrate the validity of the 1.15-1.22 micron region to recover accurate
stellar metallicities from Solar and metal-poor (one tenth Solar) spectral
templates. From tests at spectral resolving powers of four and ten thousand, we
require continuum signal-to-noise ratios in excess of 50 (per two-pixel
resolution element) to recover the input metallicity to within 0.1 dex. We
highlight the potential of direct estimates of stellar metallicites (over the
range -1<[Fe/H]<0) of red giants with the E-ELT, reaching out to distances of
~5 Mpc for stars near the tip of the red giant branch. The same simulations are
also used to illustrate the potential for quantitative spectroscopy of red
supergiants beyond the Local Volume to tens of Mpc. Calcium triplet
observations in the I-band are also simulated to provide a comparison with
contemporary techniques. Assuming the EAGLE instrument parameters and simulated
performances from adaptive optics, the J-band method is more sensitive in terms
of recovering metallicity estimates for a given target. This appears very
promising for ELT studies of red giants and supergiants, offering a direct
metallicity tracer at a wavelength which is less afffected by extinction than
shortward diagnostics and, via adaptive optics, with better image quality.
