theta = tan^-1 x if and only if -pi/2 < theta < pi/2 and tan theta = x
cos(2theta) = 2cos^2 theta -1
There are several ways to find cos^2 theta for tan theta = x.
Here are two:
Method 1: Sketch a triangle
You can sketch a right triangle with one angle theta. Label the side opposite theta as length x and the side adjacent has length 1, so the hypotenuse has length sqrt(1+x^2)
We can see that cos theta = 1/sqrt(1+x^2)
Method 2: Use a trigonometric identity
Recall that tan^2 theta +1 = sec^2 theta = 1/cos^2 theta.
So x^2+1 = 1/cos^2 theta.
Using either method we continue
cos^2 theta = 1/(1+x^2), so
cos(2theta) = 2/(1+x^2) - 1
= (1-x^2)/(1+x^2).
And we have
cos(2tan^-1 x) = (1-x^2)/(1+x^2)
