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If $\alpha+\beta+\gamma=\pi$ then $\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$.

If $\alpha+\beta+\gamma=\pi$ then $\sin(2\alpha)+\sin(2\beta)+\sin(2\gamma) = 4\sin\alpha\sin\beta\sin\gamma$.

These get referred to as "conditional identities" in books on trigonometry.

Are there any out there in published sources where the "condition" is something other than that the sum of three angles or (to be moderately daring?) the sum of several angles, is $\pi$? How many relatively basic ones (as opposed to complicated ones derived from these and others like them) are in somewhat standard sources besides these two? Any at all?

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Here is an identity I made up. Suppose that $$a+b+c+d=2\pi.$$ Then $$4\cos(a)\cos(b)\cos(c)\cos(d)-4\sin(a)\sin(b)\sin(c)\sin(d)$$ $$=\cos(2a)+\cos(2b)+\cos(2c)+\cos(2d).$$

Are these the type of things that you are looking for?

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I'm not sure this one isn't merely one of the "derived" sort I mentioned. I know lots of those. For example, if $\alpha+\beta+\gamma+\delta+\varepsilon=\pi$ then $\sin(2\alpha)+\cdots+\sin(2\varepsilon)$ $=4\underbrace{\sin\alpha\sin\beta\sin\gamma}_{\text{sines}}\ \underbrace{\cos\delta\cos\varepsilon}_{\text{cosines}}+\{\text{9 other such terms}\}$ ${}-8\sin\alpha\sin\beta\sin\gamma\sin\delta\sin\varepsilon$.${}\qquad{}$ –  Michael Hardy Dec 26 '12 at 1:20
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