How would we know if a distant planet had an Earth-like atmosphere?
The basic idea is that if a biosphere exists that is similar to the Earth, dominated by plants respiring according to the photosynthesis process, atmospheric carbon dioxide will be steadily converted into oxygen, and these oxygen atoms will produce an ozone layer which emits certain unique spectral lines in the spectrum of the planet. With spectroscopic techniques, we should be able to see the 'ozone line' in the spectra of planets orbiting nearby stars. We can also detect carbon dioxide absorption features and even methane and ammonia. Together, these lines will tell us the principle atmospheric constituents, and whether the planet is Earth-like or not. If you detect ozone, there MUST be a biosphere, because free oxygen cannot exist in an atmosphere without constant replenishment. The only chemical process we know to replenish oxygen on a planetary scale is by the photosynthesis process.