Simplifying tan(2arccotx) - Peter's Question at Yahoo Answers

[MarkFL]

Here is the question:

How to simplify tan(2arccot x)?

How do you to simplify tan(2arccotx) as much as possible?

(using trigonometric identities)

thank you!

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Peter,

We are given to simplify:

\(\displaystyle y=\tan\left(2\cot^{-1}(x) \right)\)

Let's first apply the double-angle identity for the tangent function, which is:

\(\displaystyle \tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}\)

and we obtain:

\(\displaystyle y=\frac{2\tan\left(\cot^{-1}(x) \right)}{1-\tan^2\left(\cot^{-1}(x) \right)}\)

Next, we may apply the identity:

\(\displaystyle \cot^{-1}(x)=\tan^{-1}\left(\frac{1}{x} \right)\)

and we obtain:

\(\displaystyle y=\frac{2\tan\left(\tan^{-1}\left(\frac{1}{x} \right) \right)}{1-\tan^2\left(\tan^{-1}\left(\frac{1}{x} \right) \right)}\)

This reduces to:

\(\displaystyle y=\frac{\dfrac{2}{x}}{1-\left(\dfrac{1}{x} \right)^2}\)

Multiplying the right side by \(\displaystyle 1=\frac{x^2}{x^2}\) we get:

\(\displaystyle y=\frac{2x}{x^2-1}\)