# Why do the two methods give different answers for the Fourier transform of $Y = \cos(\omega_0 t + \phi)$?

Why do the following two methods give different answers (or are they the same) for the Fourier transform of $$Y = \cos(\omega_0 t + \phi)$$, with respect to $$t \to \omega$$ ?

Using the Dirac impulse sifting property $$f(x) \delta(x-a) = f(a)\delta(x-a)$$ you can also verify that the second method produces the same output as the first method.
• you can't go from method-2 into method-1. Take for example $$f(x) = 5 \delta(x-1) + 2 \delta(x-2)$$ this has infinitely many possible equivalents such as $$f_1(x) = (x+4)\delta(x-1) + x \delta(x-2)$$ or just another $$f_2(x) = (2x+3)\delta(x-1) + (3x-4) \delta(x-2)$$; after a generalization, as you can see there are infinite number of possible eqivalents... – Fat32 Apr 11 at 10:08