Question

Simplify

Answer 1

\[\cos(2\tan^{-1} x)\]

Answer 2

\[\large\rm \cos\left[2\color{orangered}{\arctan x}\right]\]So let's take a look at this arctangent first.

Answer 3

Okay

Answer 4

Let's call it something, like theta.
Inverse trig functions give us angles, so this seems natural to do so.\[\large\rm \color{orangered}{\arctan x=\theta}\qquad\to\qquad \tan \theta=x\]From there we can rewrite it in terms of tangent,
is that step confusing?

Answer 5

Yea a little. What happened to arctanx?

Answer 6

We inverted it.
You can apply this step if it makes more sense to you,\[\large\rm \arctan x=\theta\]Taking tangent of both sides,\[\large\rm \tan\left(\arctan x\right)=\tan \theta\]Since our tangent and inverse tangent are inverses of one another, they simple "undo" one another.\[\large\rm x=\tan \theta\]

It might make sense if you think an example using multiplication and division (inverses).\[\large\rm 5\frac{x}{5}=multiply 5(divide5(x))\]These operations are inverses of one another.
They undo each other. Or in this case, they "cancel out" , ya?

Answer 7

It's usually more fluid to get used to doing it this way though,
where you rewrite arctan as tan, and switch the x and theta.

Answer 8

Because that's what ends up happening when you take inverse.
The "x" and "y" trade places and the function becomes the inverse of itself.

Answer 9

Oh i get it. But where did the theta come from?

Answer 10

We're taking this,\[\large\rm \arctan x\]and we're naming it something.
`It is an angle`, so it makes sense to call it theta or something.
But you can call it whatever you like,\[\large\rm \arctan x=u\]

Answer 11

Oooh. Okay i get it

Answer 12

So we have a tangent,\[\large\rm \tan \theta=x\]We know this about tangent, yes?\[\large\rm \tan \theta=\frac{opposite}{adjacent}\]So maybe we can draw a triangle and place this x somewhere.

Answer 13

So how can we label the sides? Any ideas?

Answer 14

Uh... No idea. :/

Answer 15

Well.. we only have a single x...
that doesn't give us two sides to label.
Maybe we could write it over 1.\[\large\rm \tan \theta=\frac{x}{1}=\frac{opposite}{adjacent}\]

Answer 16

Oh okay

Answer 17

So label the triangle :U DO ITTTT!

Answer 18

Sorry the website it being glitchy