It is well known (?) that if $\alpha+\beta+\gamma=\pi$ then $4\sin\alpha\sin\beta\sin\gamma = \sin(2\alpha)+\sin(2\beta)+\sin(2\gamma)$ (I think I've seen it in some late-19th-century books, and I read somewhere on the internet (therefore it's true!! right?) that it has repeatedly appeared on the...
I think this question deserves a more descriptive title than just "Trigonometric identity".
The title "Trigonometric identity: If $\alpha+\beta+\gamma=\pi$ then $4\sin^2\alpha\sin^2\beta\sin^2\gamma= (\sin\alpha+\sin\beta+\sin\gamma)(\sin\alpha+\sin\beta-\sin\gamma)(\sin\alpha-\sin\beta+\sin\gamma)(-\sin\alpha+\sin\beta+\sin\gamma)$" is 231 characters, limit is 150.
"If $\alpha+\beta+\gamma=\pi$ then $4\sin^2\alpha\sin^2\beta\sin^2\gamma=(\sin\alpha+\sin\beta+\sin\gamma) (\sin\alpha+\sin\beta-\sin\gamma)(\sin\alpha-\sin\beta+\sin\gamma)(-\sin\alpha+\sin\beta+\sin\gamma)$" is still 209
The RHS alone is 136 characters: $(\sin\alpha+\sin\beta+\sin\gamma)(\sin\alpha+\sin\beta-\sin\gamma)(\sin\alpha-\sin\beta+\sin\gamma)(-\sin\alpha+\sin\beta+\sin\gamma)$
What is your suggestion for better title?
Identity for $4\sin^2\alpha\sin^2\beta\sin^2\gamma$ where $\alpha+\beta+\gamma=\pi$
Identity for $4\sin^2\alpha\sin^2\beta\sin^2\gamma$ expressed as product of terms $(\pm\sin\alpha\pm\sin\beta\pm\sin\gamma)$
Does some of the above two suggestions seem reasonable?
How can you simplify $\cos^6x+\sin^6x$ to $1−3\sin^2x\cos^2x$?
I know that $\cos^2x+\sin^2x=1$. I thought about trying to use $(\cos^2x+\sin^2x)^3=1$, but that did not lead me anywhere.
I found some other questions about the same expression, but they simplify this to another form: Finding $\si...
I wondered how the questions could be salvaged. (So that we do not lose answers posted there.)
One possibility would be to find a duplicate and merge (so that the answers would be moved there). But I did not find a duplicate.
The other possibility would be improve the question.
I know that it is not ideal if "fake context" is added by somebody else than the OP. But seeing that there are already two delete votes, if we want to do something with the post, we are probably running out of time.
So I have edited it. Further improvements are welcome.
@MartinSleziak Or another possibility would be to make a completely new post (with enough context) and then ask moderators to merge/move answers. But let's wait to see whether the above post really gets deleted.
@DanielFischer I am not sure whether you read the above comments. I think that the question was closed deservedly. (And my attempt to add some context was not sufficient to get it reopened from the review.)
Would it be ok if I post a new question about the same identity (and with sufficient context) and then flag for merging?
Main goal is not to lose the answers. (I think that question about this and also answers could be useful for some people.)
As I have mentioned above, I was not able to find a duplicate. (If such question already exists on the site, it would be a better target for merging.)
@MartinSleziak Since I don't need to wait for four other dupe-votes (hmm, wonder when we'll get the first fake-proofs gold badge), that's not necessary.
@DanielFischer Still I thought that it might be reasonable to leave it here. There are users who enjoy searching for questions. And, moreover, if somebody else tries to search, we might find a few more duplicates.
I didn't know whether you were already looking, and in case you hadn't been, wanted to un-ping, to not cause pointless work for you. Could have edited the @Martin out, but deleting is quicker.
Question is :
Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:
$X$ is not connected.
$X$ is not compact
$X$ is not homeomorphic to a subset of $\mathbb{R}$
None of the above.
I guess first two options are fa...
Given that $X$ is a topological space with infinite cardinality and $X$ is homeomorphic to $X\times X$.
(A) $X$ cannot be connected.
(B) $X$ cannot be compact.
(C) $X$ cannot be homeomorphic to a subspace of $\mathbb R$.
(D) None of the above.
If I take $\mathbb Q$...
@PraphullaKoushik The time is not the main thing in order to decide which of the two question should be duplicate target, see meta. Of course, if you disagree, feel free to vote to close the other one. — Martin Sleziak59 secs ago
67754 requires non-discrete space, 722412 requires connected space, the only requirement in 1384455 is that the space is infinite.
As far the answers go, in my opinion the simples answer is the space of the form $X=A^{\mathbb N}$ for some $A$ (countable power). Both 722412 and 1384455 have such answer.
