MHB 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:

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

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

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

and we obtain:

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

Next, we may apply the identity:

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

and we obtain:

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

This reduces to:

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

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

$$y=\frac{2x}{x^2-1}$$