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Explain notation for sum of cos terms [Pending...]

My question

Please explain the meaning of the notation used in Wikipedia for a trig identity involving the product and sum of cosines where more than two angles are involved. What does the expanded identity look like for the number of angles equals, say, four or five? I have tried substituting for n equals two or three but do not understand the notation. I am attaching an image of the identity copied from Wikipedia.
Thank you.

Relevant page

Product to Sum Trigonometric Identities

What I've done so far

general

X

Please explain the meaning of the notation used in Wikipedia for a trig identity involving the product and sum of cosines where more than two angles are involved.  What does the expanded identity look like for the number of angles equals, say, four or five?  I have tried substituting for n equals two or three but do not understand the notation. I am attaching an image of the identity copied from Wikipedia.
Thank you.
Relevant page

<a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities">Product to Sum Trigonometric Identities</a>

What I've done so far

general

Re: Explain notation for sum of cos terms

@gtr109 OK, let's start with the left hand side of the equation.

Can you expand out the first 4 terms, that is `prod_(k=1)^4 cos theta_k`?

BTW - this is the formula from Wikipedia entered using the forum's math entry system:
`prod_(k=1)^n cos theta_k ` ` =1/2^n sum_(e in S) cos(e_1 theta_1 + ... + e_n theta_n)` where `S={1,-1}^n`

X

@gtr109 OK, let's start with the left hand side of the equation.

Can you expand out the first 4 terms, that is `prod_(k=1)^4 cos theta_k`?

BTW - this is the formula from Wikipedia entered using the forum's math entry system:
`prod_(k=1)^n cos theta_k ` ` =1/2^n sum_(e in S) cos(e_1 theta_1 + ... + e_n theta_n)`  where `S={1,-1}^n`

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