Trigonometric Identity

**ematrix** · November 11th 2011, 06:45 PM

Prove the following identity.
cos2pi/7 +cos4pi/7 + cos8pi/7 = -1/2

**mathbyte** · November 11th 2011, 09:04 PM

We have
cos(2pi/7) +cos(4pi/7) + cos(8pi/7) = -1/2

The best way to prove this will be to simplify the left side. To do that we need to be able to try and collect like terms.

There is a trigonometric identity that says:
sin(x)cos(y) = (1/2)[(sin(x+y)+sin(x-y)]

Let's multiply both sides of the equation by sin(pi/7):
sin(pi/7)cos(2pi/7) + sin(pi/7)cos(4pi/7) + sin(pi/7)cos(6pi/7) = -1/2

Now, we apply the identity above:
(1/2)[sin(pi/7+2pi/7) + sin(pi/7-2pi/7)] + (1/2)[sin(pi/7+4pi/7) + sin(pi/7-4pi/7)]
+ (1/2)[sin(pi/7+8pi/7) + sin(pi/7-8pi/7)] = -(1/2)sin(pi/7)

We can multiply both sides by 2 and simplify the terms a bit:
[sin(3pi/7) + sin(-pi/7)] + [sin(5pi/7) + sin(-3pi/7)] + [sin(9pi/7) + sin(-7pi/7)] = -sin(pi/7)

Further simplifying:
sin(3pi/7) - sin(pi/7) + sin(5pi/7) - sin(3pi/7) + sin(9pi/7) - sin(7pi/7) = -sin(pi/7)

Now, we can collect like terms:
sin(5pi/7) + sin(9pi/7) - sin(7pi/7) = 0

But sin(7pi/7) = 0:
sin(5pi/7) + sin(9pi/7) = 0

At this point you can probably inference whether the identity will be proved, or not.

By the way, was the problem stated/copied correctly? I find myself wondering of the last term of the left half of the equation should be cos(6pi/7), not cos(8pi/7).

Hope this helps!

**Amer** · November 11th 2011, 11:05 PM

sin(5pi/7) + sin(9pi/7) = 0 this is true, write 5pi/7 = pi - 2pi/7 , 9pi/7 = pi + 2pi/7 and use the identities

sin(a+b), sin(a-b)

**Opalg** · November 12th 2011, 01:34 AM

Originally Posted by ematrix

Prove the following identity.
cos2pi/7 +cos4pi/7 + cos8pi/7 = -1/2

Here's a picture to illustrate this problem. It shows seven spots equally spaced round the unit circle. The black spot is at (1,0). The x-coordinates of the red spots are $\cos2\pi/7$ , $\cos4\pi/7$ and $\cos8\pi/7$ , as also are the x-coordinates of the blue spots. The centre of mass of the seven spots is the origin. So the sum of the x-coordinates of the red spots plus the x-coordinates of the blue spots plus 1 (the x-coordinate of the black spot) is equal to 0. It follows that the sum of the x-coordinates of the red (or the blue) spots is –1/2.

That is an "applied" approach to the problem. You probably want a pure mathematical proof. There are several possible methods for this. One is to look at solutions of the equation $\cos 7x = 1.$ Another is to use complex numbers: if $z=e^{2\pi i/7}$ then $z^7=1.$ But I think that the applied approach is the one that gives the most genuine insight as to why the result is true.