I am studying the Fourier transform and trying to understand the following equation

w is frequency..

t is time..

A(w) is a a signal amplified function

delay(w) is a time delay function

Trig Identity: sin( a + b ) = sin(a)cos(b) + sin(b)cos(a)

Therefore:

A(w)sin((wt + delay(w))) = A(w)cos(delay(w))sin(wt) + A(w)sin(delay(w))cos(wt) -

cos(a) = sin(a + pi/2) The cosine is just a time-advanced sine,

it follows that the response to the input cos(wt) is just:

A(w)cos(delay(w))cos(wt) - A(w)sin(delay(w))sin(wt) -

I don't understand how eq.1 get converted into eq.2... Any insights would be much appreciated.

Thanks.

w is frequency..

t is time..

A(w) is a a signal amplified function

delay(w) is a time delay function

Trig Identity: sin( a + b ) = sin(a)cos(b) + sin(b)cos(a)

Therefore:

A(w)sin((wt + delay(w))) = A(w)cos(delay(w))sin(wt) + A(w)sin(delay(w))cos(wt) -

**eq.1**cos(a) = sin(a + pi/2) The cosine is just a time-advanced sine,

it follows that the response to the input cos(wt) is just:

A(w)cos(delay(w))cos(wt) - A(w)sin(delay(w))sin(wt) -

**eq.2**I don't understand how eq.1 get converted into eq.2... Any insights would be much appreciated.

Thanks.

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