#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)#