# Cot(arc tan x)

1. Aug 19, 2015

### Sobhan

• Member warned about posting with no effort shown
The answer to Cot(arc tan x)

2. Aug 19, 2015

### Mentallic

How else can you express $\cot{\theta}$? Also, arctan is a function and every function is only ever one word, so the way you should be writing this is $\cot\left(\arctan(x)\right)$

3. Aug 19, 2015

### Sobhan

Cot(arc tan (x))?

4. Aug 19, 2015

### Mentallic

I was simply explaining that functions such as cot(x), arctan(x), sin(x), ln(x) etc. are always written without any spaces.

arc tan (x) is wrong
arctan(x) is right

cot(arc tan (x)) is wrong
cot(arctan(x)) is right

Ok, so now that we have that out of the way, let's go back to the problem at hand. You want to simplify $\cot(\arctan(x))$. Firstly, how else can you express $\cot(x)$? That function has a very obvious relationship with tan(x) because it was defined that way. What is it?

5. Aug 19, 2015

### BvU

Strange, I suggested you should make a little drawing to understand the question, but it seems to have gone lost in transmission

6. Aug 19, 2015

### HallsofIvy

Staff Emeritus
Do you not know that, by definition, $cot(\theta)= \frac{1}{tan(\theta)}$?

7. Aug 19, 2015

### Sobhan

is it x^-1?

8. Aug 19, 2015

### Mentallic

9. Aug 19, 2015

### Sobhan

thanks for making me think.

10. Aug 21, 2015

### Sobhan

What is cot(0.5×arctan(x))

11. Aug 21, 2015

### Mentallic

Search for how else you can represent
$$\tan(\theta / 2)$$

and decide whether this can help you. Sometimes expressions can't be simplified further, but they can be transformed regardless.

To help you on this journey, you might also need to figure out how to simplify $\sin(\arctan(x))$ and $\cos(\arctan(x))$. The way you can do this is by drawing a right triangle, and then let one angle be $\theta$. This angle will actually be $\arctan(x)$ though, which means that $\theta=\arctan(x)$ hence $\tan(\theta)=x$. If $\tan(\theta)=x$ then can you label every side of the triangle in terms of x?

12. Aug 21, 2015

### Sobhan

I have reached a equation to express cot(x) by cot(2x) and it is not an easy one the equation is this (2cot(2x)+(-)(4cot(2x)^2+4)^0.5))×2^-1
Is this right?

13. Aug 21, 2015

### BvU

Did you make a drawing again ?

14. Aug 21, 2015

### Mentallic

Did you derive that for yourself or take it from somewhere else? It also has an obvious common factor of 2 in the numerator and denominator.

Sorry BvU, I forgot all about your drawing further up. It could've saved me quite a bit of explaining in my last post if I just referred to your diagram.

15. Aug 21, 2015

### BvU

Is it clear to all that if $\cot(\arctan x)$ is a very simple expression, then perhaps $\cot ({1\over 2} \arctan x)$ mght also be a simple expression ?

 LaTeX justifiably does not recognize atan as a mathematical function name. It indeed should be arctan (no space) and not atan as I first typed ... .

 woops, all wrong, sorry . Wake up first, then check alerts! Not the other way around.

Last edited: Aug 21, 2015
16. Aug 21, 2015

### Sobhan

I got that from this : cot(2x)=0.5(cot(x)-tg(x))

17. Aug 21, 2015

### Mentallic

It's definitely been a while since I've seen tg.

Ok awesome, so can you finish it off?