Fourier Integrals and Division

[Yosty22]

Homework Statement

(a) Find the Fourier transform f(ω) of: f(x) = cos(x) between -pi/2 and pi/2
(b) Find the Fourier transform g(ω) of: g(x) = sin(x) between = -pi/2 and pi/2
(c) Without doing any integration, determine f(ω)/g(ω) and explain why it is so

Homework Equations

f(ω) = ∫f(x)e^-iωx dx

The Attempt at a Solution

I was able to do parts (a) and (b) and verified my answers, however part (c) is giving me some problems. The division is straightforward for the two transformed equations. When I do the division, I get f(ω)/g(ω) = -1/(iω). I have verified this with fellow students as well, and we have all gotten the same thing. I am just confused as to the why . I feel like it might be some identity of Fourier integrals, but I cannot find it. I have looked through my textbook and I have been looking online, but I cannot really understand exactly why I get the answer I do.

Any help would be greatly appreciated.

 

[vela]

Hint: What linear operator can you apply to g(x) to get f(x)?

 

[vanhees71]

Even easier: What linear operator can you use to come from $f(x)$ to $g(x)$.