Write the given expression as an algebraic expression in $x$: $\tan(2 \cos^{−1} x)$. OK I first assigned $\theta = \cos^{−1} x$
and so $\cos(\theta) = x$
drew a triangle where adj = $x$  hyp = $1$ and opp = $\sqrt{1-x^2}$ by pyth theorem.
Then $\tan(2\cos^{−1} x) = \tan(2\theta)$
and I used the double angle formula for tangent so
$\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$
Then used the triangle: $\tan(\theta) = \frac{\sqrt{1-x^2}}{x}$
but my answer is 
The computer states I am wrong. Am I setting the problem up wrong? 

Your setup is correct, but there seems to be an error in your algebra after you obtain $\tan \theta = \frac{\sqrt{1-x^2}}{x}.$  You should have $$\tan 2\theta = \frac{2 \sqrt{1-x^2}/x}{1 - (1-x^2)/x^2} = \frac{2x \sqrt{1-x^2}}{2x^2 - 1}.$$
