Hello, Kaitosan!

I saw these derived many year ago and the approach is still with me.

We use these Compound Angle Identities:

. .

. .

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Product-to-Sum Identities

. .

We have: .

Derivation of [1]

Add

And we have: .

Derivation of [2]

Subtract

And we have: .

Derivation of [3]

We have: .

Add

And we have: .

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Sum-to-Product Identities

. .

These can be derived from the previous four identities with a substitution.

Let: .

From [3] we have: .

Substitute [1] and [2]: .

. . . and so on.