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?
