Let #f(t)# be a real function of the proper class. By definition the Dirac distribution #delta(t)# is such that:
#< delta, f > = f(0)#
so:
#int_(-oo)^(+oo) delta(t) e^(-i omega t) dt = 1#
and if we translate the distribution:
#int_(-oo)^(+oo) delta(t-t_0) e^(-i omega t) dt = e^(-i omega t_0)#
So the Fourier transform of #delta(t-t_0)# is # e^(-i omega t_0)#.
By the symmetry property of the transform if we let: #t_0 = -omega_0# then the Fourier transform of #e^(-i omega_0 t)# is #2pidelta(omega-omega_0)#
But using the expression of the cosine as:
#cos (omega_0t) = (e^(i omega_0 t) + e^(-i omega_0 t))/2#
and the linearity of the transform then:
#cos (omega_0t) harr pi(delta(omega-omega_0) + delta(omega+omega_0))#
