I think you are on the right track
simplify cos(2*arctan(x)) - Wolfram|Alpha
I think you are on the right track
simplify cos(2*arctan(x)) - Wolfram|Alpha
And, if , then .
We can represent that by a right triangle having angle , "opposite side" x, and "near side" 1. By the Pythagorean theorem the hypotenuse has length . So what are and ?