Trigonometry and method of difference

[gaobo9109]

Homework Statement

Given that 1+2cosx+2cos2x+2cos3x + ....+2cosnx = sin(x+1/2)x cosec(x/2)
Prove that 1-2cosx+2cos2x-2cos3x+....+2cos8x = cos(17/2)x sec(x/2)

Homework Equations

The Attempt at a Solution

1+2cosx+2cos2x+2cos3x + ....+2cos8x = sin(8+1/2)x cosec(x/2)

4cosx+4cos3x+4cos5x... = ?

I don't know how to continue. Or is there another way?

 

[NascentOxygen]

1+2cosx+2cos2x+2cos3x + ....+ 2cos8x = ... eqn(1)
Substitute 2x for x, and you can say
1 + 2cos2x + 2 cos4x + 2cos6x + 2cos8x = ... eqn(2)

If you subtract double eqn(2) from eqn(1) you'll as good as have the left hand side. I don't know whether this will get you anywhere.

Also, fix up the x that should be n in your first line.