Math Help - [SOLVED] What is the identity used here?

1. [SOLVED] What is the identity used here?

This is part of a larger example problem that I came across in reviewing. I can't for the life of me remember the identity or principle that would get one from the first form to the second in one step. Maybe it is not one step -- the authors of the textbook eliminated what they assumed one would see automatically? I realize this may be a stupid question, but it is driving me crazy. Anyone?

$\cos{(\pi-2\theta)} = \frac{20-x}{x}$

$\cos{(2\theta)} = \frac{x-20}{x}$

2. Hello sinewave85
Originally Posted by sinewave85
This is part of a larger example problem that I came across in reviewing. I can't for the life of me remember the identity or principle that would get one from the first form to the second in one step. Maybe it is not one step -- the authors of the textbook eliminated what they assumed one would see automatically? I realize this may be a stupid question, but it is driving me crazy. Anyone?

$\cos{(\pi-2\theta)} = \frac{20-x}{x}$

$\cos{(2\theta)} = \frac{x-20}{x}$
For any angle $\theta, \cos(\pi-\theta) =-\cos(\theta)$

So $\cos(\pi-2\theta) = \frac{20-x}{x}$

$\Rightarrow -\cos(2\theta) = \frac{20-x}{x}$

$\Rightarrow\cos(2\theta) = -\frac{20-x}{x} = \frac{x-20}{x}$

For any angle $\theta, \cos(\pi-\theta) =-\cos(\theta)$
Thanks, Grandad! I knew it had to be something simple (kind of embarased that it was that basic), but unfortunately those are sometimes the hardest things for me to remember. You saved me a lot of searching and headbanging.

P.S. And thanks for writing out all of the steps so nicely! It helps to see each little adjustment.