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Edit - question modified

I started a question on the product of a sum of sines producing unity; however, I realized that the question was too specific. I would like to make the question more generic; so I am seeking a product of sines or sum of sines which produces a constant.

This is one possible scenario which I had previously equaling one: $$(A\sin(w_1*t)+B\cos(w_2*t)+...)*(Z\sin(w_n*t)+Y\cos(w_{n+1}*t)+...)=\Omega$$

In my previous post, someone suggested:

$$(\cos(w*t)+i*\sin(w*t))(\cos(w*t)-i*\sin(w*t))=1$$

but the imaginary number is undesirable. Normally, I take an imaginary number as a phase shift by $90$ degrees but I cannot find a combination of this equation, where $i$ is replaced by a sine wave with a $90$ degree phase shift, that produces a constant.

I've scoured trig and complex number text books and haven't found much. Similar to this question, I found this relation:

$$\prod_{k=1}^{n} \cos\left(\frac{k*\pi }{2*n+1}\right)=\frac{1}{2^{n}}$$

but this deals with sines at specific points not those which are a function of time (e.g. $\sin(w*t)$).

I also realized that a product of sines produces spectra at the sum and the difference of the frequencies:

$$\sin(a)*\sin(b)=\frac{\sin(a+b)+\sin(a-b)}{2}$$

So I set out to find a series of frequencies that would add destructively everywhere but at DC (i.e freq $=0$ ); however, I had no luck.

I am wondering if anyone has any ideas?

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  • $\begingroup$ $(\cos\theta+i\sin\theta)(\cos\theta-i\sin\theta)=1$ $\endgroup$ – user Apr 24 '18 at 14:41
  • $\begingroup$ Lol. Yes you are right. And $\i\sin\theta=\cos(theta+\pi/2)$ $\endgroup$ – Jordan McBain Apr 24 '18 at 14:48

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