You can get the phase from the output of fft()
Fs = 1000;
t = 0:1/Fs:1-1/Fs;
x = cos(2*pi*100*t-pi/4)+cos(2*pi*200*t-pi/2);
xdft = fft(x);
angle([xdft(101) xdft(201)])
but you have to keep in mind that:
1.) Phase is only defined modulo 2*pi
2.) Additive noise (if it is white) occurs at all frequencies so it will affect the phase.
You can see from the example above, the phase is correctly identified at 100 and 200 Hz.
Also, I don't understand your statement:
"I have a random signal and I need to remove only one frequency component from the signal. If I don't have the time-phase shift of that frequency component, then I cant remove it from the signal. How do I get the phase angle from FFT?"
If you really are just removing one sinusoidal component, why do you need the phase, you just need to know the DFT bin of the "positive" and "negative" frequency component.
Remove 200 Hz.
N = length(xdft);
conjbin = N-201+2;
ydft = xdft;
ydft([201 conjbin]) = 0;
y = ifft(ydft);
plot(y) % 200 Hz is removed
