#help me
magic roverBOT
daring saffron
haven't tried the math at all cus I'm on my phone so idk if this will help but imma guess express the summands in terms of e cus euler's identity and tagged calculus and roots of unity, also maybe induction but good chance it doesn't require induction 
daring saffron
OK been trying it for a bit but can't get it to telescope or each summands to have a common denominator
Roots of unity just has to come in at some point tho
young kraken
facts
daring saffron
Tryna use the fact that all the roots of x^n - 1 besides x = 1 (I. E. the k = 0 case) satisfy the equation x^(n - 1)+ x^(n - 2) +... + 1 = 0
So in particular sum of those roots is (-1)^n and similar formulas for all the other symmetric sums
daring saffron
OK so this is what I mean with the symmetric sums. Basically you can get types of sums related to the roots in terms of the coefficients of the polynomial: https://artofproblemsolving.com/wiki/index.php/Vieta's_formulas
So like for instance (x - 1)(x - 2) = x^2 - 3x + 2
If we didn't know the factorization
We could still know the sum of the roots from the 2nd coefficient, which would be - 3/(-1)^1
And the product of the roots would be 2
Then these are the roots of unity https://brilliant.org/wiki/roots-of-unity/
The solutions to the equation x^n - 1 = 0
Which factors as (x - 1)(x^(n - 1) +..... + 1)
The roots are of the form e^(2 pi k i/n) for k = 1,....n - 1
k = 0 is just 1, so all the other ones make up the roots of the polynomial x^(n - 1) +... + 1
And we'll what's noice about that is then all the symmetric sums are just $\pm 1$
sinful pythonBOT
992qqoloy
daring saffron
Then sin(x) = (e^(ix) - e(-ix))/(2i)
Squaring for this instance you get [e^(2pi i k/n) + e^(-2pi i k/n) - 2] /(-4)
So that looks promising but idk where to go from there 
Bleh if you had the sin^2 in numerator instead it'd be so simple
sinful pythonBOT
aryanh143
short kraken
I think I made huge blunders
daring saffron
You're close, should be - instead of + for sine
And dividing by 2i so squared makes -4
short kraken
I didn't get that help me in further solution
daring saffron
yeah 😐
short kraken
Sorry
daring saffron
Np
Tryna common denominator thing now
Like I wonder if the product of sines or cosines of roots of unity has a nice formula
short kraken
And dividing by 2i so squared makes -4
I don't know why but I didn't understand things from this
daring saffron
Like the formula for sin(x) is (e^(ix) - e^(-ix)) /(2i)
short kraken
**aryanh143**
What I do wrong in this?
daring saffron
But you used the formula for cos when evaluating sin^2(x)
OK Jesus christ so I looked up the solution
None of them are remotely simple 
https://math.stackexchange.com/questions/544228/finite-sum-sum-limits-k-1m-1-frac1-sin2-frack-pim if you're curious
Dunno how you'd ever get it with just a basic knowledge of Calc and euler's identity
I mean you can understand the solution with those two
But they require tricks they dont teach in normal classes I think
short kraken
A very big thanks
daring saffron
np
sinful pythonBOT
aryanh143
