  This article constructs flat-sky approximations in a controlled way in the
context of the cosmic microwave background observations for the computation of
both spectra and bispectra. For angular spectra, it is explicitly shown that
there exists a whole family of flat-sky approximations of similar accuracy for
which the expression and amplitude of next to leading order terms can be
explicitly computed. It is noted that in this context two limiting cases can be
encountered for which the expressions can be further simplified. They
correspond to cases where either the sources are localized in a narrow region
(thin-shell approximation) or are slowly varying over a large distance (which
leads to the so-called Limber approximation). Applying this to the calculation
of the spectra it is shown that, as long as the late integrated Sachs-Wolfe
contribution is neglected, the flat-sky approximation at leading order is
accurate at 1% level for any multipole. Generalization of this construction
scheme to the bispectra led to the introduction of an alternative description
of the bispectra for which the flat-sky approximation is well controlled. This
is not the case for the usual description of the bispectrum in terms of reduced
bispectrum for which a flat-sky approximation is proposed but the
next-to-leading order terms of which remain obscure.
