Proving inverse functions identity

Hello everybody and thanks for reading.

I have been trying to solve the following problem without success, so I am looking for some advices..

I have the following equation

arccos(x) = arctan( root(1 - x^2) / x)

So I have to prove if that 'identity' is valid and say for what values of x is valid.

Even when I am able to construct the triangle from the data, I don't know how to find the valid values for x.

So far I have constructed a triangle with opposite root(1-x^2), adjacent x and hypotenuse 1.

I also restricted the domain so 1-x^2 >= 0, x != 0 and -1 <= x <= 1 (for the root, denominator, and arccos respectively).

I also tried to prove the identity using the Pythagorean theorem from the data of the arctan ( opposite and adjacent), and I get that the hypotenuse is 1, which is the same the arccos says.

But I have seen the plot of both functions and I realized they are only equal for 0 < x <= 1 and I don't know how I could get there. Intersecting the restrictions I said I only get [-1,1] - {0}.

I have even thought I should intersect the ranges of those functions, but I don't think that is valid.

Thanks in advance and sorry for the long post

Re: Proving inverse functions identity

Quote:

Originally Posted by

**Danielc** I have the following equation

arccos(x) = arctan( root(1 - x^2) / x)

So I have to prove if that 'identity' is valid and say for what values of x is valid.

Even when I am able to construct the triangle from the data, I don't know how to find the valid values for x.

So far I have constructed a triangle with opposite root(1-x^2), adjacent x and hypotenuse 1.

Let , so .

Thus .

What don't you follow?

Re: Proving inverse functions identity

Yes, that is what I did.. But if you try the identity for, say, -1, it is not true... So what I need to do is to find (hopefully algebraically) the values that make that identity true.

I constructed the triangle doing the same as you did, but I don't know how should i proceed.

Thank you very much

Re: Proving inverse functions identity

Re: Proving inverse functions identity

(deleted due to previous post deletion)

Re: Proving inverse functions identity

How can posts be deleted? There is no delete button at the top of my edit window(Talking)

Re: Proving inverse functions identity

Quote:

Originally Posted by

**MaxJasper** How can posts be deleted?

EDIT your post.

At the very top of the edit box there is a delete option.

Re: Proving inverse functions identity

Hi and thanks for your answer.

Yes, I can see where it is valid graphically, but I would like to know how to get to that conclusion algebraically

Thank you very much

1 Attachment(s)

Re: Proving inverse functions identity

If you are dealing with real numbers only then :

Hence we investigate:

http://mathhelpforum.com/attachment....1&d=1347575251

Re: Proving inverse functions identity

Let denote . We have the following facts.

If , then (1)

If , then (2)

From (2), if , then (3)

The function arctan is the inverse of tan on (4)

(5)

From (3) and (4), if , then , so (6)

So, if , then

by (1)

by (3).

Therefore,

by (6). For we simialrly have and , so .

Re: Proving inverse functions identity

Thank you very much for your answers.

I wanted something like what emakarov posted, so it helped a lot. I tried to put it in simple terms so concluded:

- tan (theta) = opposite / adjacent, but as we are talking about lengths, negative values for any of both have no sense. In this case, x was in the denominator, si if it is negative it has no sense.

- So tan(theta) = sin(theta) / cos(theta) . As the numerator is always non-negative, theta must be an angle between [0, PI] so its sin is positive. Then, for theta to be positive, x >= 0, but as it can't be 0, it must be between (0, 1], considering the restrictions on the domain.

Thank you very much