Need help on proving this equation.?

Question

I'll be really grateful if someone can help me with this!

sinπ/7+sin2π/7+sin3π/7=1/2cotπ/14

Sorry that the other part of the cotangent wasn't visible; I wrote it but it automatically put three dots, well anyways, heres the correct version:

sin(pi/7) + sin(2pi/7) + sin(3pi/7) = 1/2(cotpi/14)

Answer 1

Let s = sin(pi/7) + sin(2pi/7) + sin(3pi/7).

Multiplying by 2sin(pi/14):
2s sin(pi/14) = 2 sin(pi/14)( sin(pi/7) + sin(2pi/7) + sin(3pi/7) )
2s sin(pi/14) = 2 sin(pi/14)sin(pi/7) + 2 sin(pi/14)sin(2pi/7) + 2 sin(pi/14)sin(3pi/7)

Using the identity 2sin(a)sin(b) = cos(a - b) - cos(a + b) on each term:

2s sin(pi/14) = cos(pi/14) - cos(3pi/14)
+ cos(3pi/14) - cos(5pi/14)
+ cos(5pi/14) - cos(pi/2)
= cos(pi/14)

Dividing by 2 sin(pi/14):
s = cot(pi/14) / 2.

Answer 2

cot π is Infinity. sin of any number is always between -1 and 1, and hence the total cannot, in any case be greater than 3 or less than -3. Hence, this is confusing.

Please check and give me the entire question

Answer 3

The cotangent of pi is undefined. You haven't given us the entire equation.

Answer 4

hey, i've finally managed to prove your eqn . but i'm afraid the soln is too long to type it here . it'll take me ages. i can send you the soln part by part by mail if you wish . send me your approval through mail .

and , by the way , great question . took me ages to solve (you got one star from me!)

edit:
but i see that rackbrane has found a much better solution. it's neat and simple . i think you should follow his method , and give him the best answer as well .