Question

Write the given expression as an algebraic expression in x.
tan(2 cos^−1 x)

Answer 1

\[\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}\]if i recall correctly
your job is therefore to find \(\tan(\cos^{-1}(x)) \)

Answer 2

And how do I find that? I have no idea how to even begin this problem.

Answer 3

this is exactly like the problem that says "if the cosine is this, find the tangent"

Answer 4

easiest way is to start with a triangle, and label the adjacent side \(x\) and the hypotenuse 1 since that cosine is adjacent over hypotenuse

Answer 5

there is the angle whose cosine is \(x\) i.e. that angle is \(\cos^{-1}(x)\)

Answer 6

we want the tangent of that angle, so we need the other side of the triangle
by pythagoras it is \(\sqrt{1-x^2}\)

Answer 7

so \(\tan(\cos^{-1}(x))=\frac{\sqrt{1-x^2}}{x}\) now make the replacement in the formula i wrote above

Answer 8

since unfortunately you are asked for
\[\tan(2\cos^{-1}(x))\] not just \(\tan(\cos^{-1}(x))\)

Answer 9

So do I just divide the answer for cos^-1 by 2?

Answer 10

no use the double angle formula i wrote in the first post