[Math] Write the given expression as an algebraic expression in $x$: $\tan(2 \cos^{−1} x)$.

Question

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?

Answer 1

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 $$\tan 2\theta = \frac{2 \sqrt{1-x^2}/x}{1 - (1-x^2)/x^2} = \frac{2x \sqrt{1-x^2}}{2x^2 - 1}.$$