# Domain of identity

1. Dec 21, 2013

### NATURE.M

1. The problem statement, all variables and given/known data

Prove that there is a constant C such that
$arctan\sqrt{\frac{1-x}{1+x}}$ = C - $\frac{1}{2}arcsinx$ for all x in a certain domain. What is the largest domain on which this identity is true? What is the value of the constant C?

3. The attempt at a solution

Now I know how to prove the initial statement (showing the derivatives are equal which implies they differ by only a constant), but I wanted to verify the largest domain and the value of C.

For the largest domain on which this identity is true I obtained (-1, 1] (since arcsinx is defined on
[-1, 1], and since -1 is not allowed).

And I believe C can be any real number.

So I'd just like to verify whether or not my interpretation is correct?

2. Dec 21, 2013

### scurty

As far as I can tell, your domain is correct.

Am I right in interpreting that you are saying this equation is true for any value of C? If so, that's not true.

To solve for C, pick any value for x in the domain, plug it into the equation, and solve for C.

3. Dec 21, 2013

### NATURE.M

4. Dec 21, 2013

### scurty

Edit: Nevermind.

To make this post worthwhile, I suggest x values of 0 or 1 to solve for C.

5. Dec 21, 2013

### NATURE.M

Yeah I used x=0 and obtained C=$\pi$/4.

6. Dec 21, 2013

### vanhees71

It's all correct, but $C$ is uniquely defined by the equation. Just choose a value for $x$, for which you know the values on both sides of the equation!