Identity Proofs of Inverse Trig Functions

In summary, the conversation is about proving the identity for the derivatives of arcsin and arctan functions. There is confusion about the correctness of the derivatives and how to use them to prove the identity. Eventually, the mistake in the derivation is corrected and the conversation ends with the understanding of how to proceed with the proof.
  • #1
MHrtz
53
0

Homework Statement



Prove the Identity (show how the derivatives are the same):

arcsin ((x - 1)/(x + 1)) = 2arctan (sqr(x) - pi/2)


Homework Equations



d/dx (arcsin x) = 1/ sqr(1 - x2)

d/dx (arctan x) = 1/ (1 + x2)

All my attempts have been messy and it may be because I didn't take the derivatives properly.
I attached what I got for the derivatives for both. If it's not the right derivative than what is? If it is the right derivative, where do I go from here?
 

Attachments

  • Derivatives.jpg
    Derivatives.jpg
    21.3 KB · Views: 402
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  • #2
Hi MHrtz! :smile:

Are you sure that that identity even holds? It doesn't seem to hold in the case x=1...
 
  • #3
Are you suggesting that the derivatives of these functions are incorrect? I got them from a textbook so I assumed they were correct.
 
  • #4
Try to substitute x=1 in

arcsin ((x - 1)/(x + 1)) = 2arctan (sqr(x) - pi/2)
 
  • #5
I get different answers for both sides but for an identity this doesn't matter. What about my derivation? Is it correct?
 
  • #6
MHrtz said:
I get different answers for both sides but for an identity this doesn't matter.

:confused: An identity means that something holds for all x, no?

What about my derivation? Is it correct?

Yes, I think that is correct.
 
  • #7
I just realized I wrote the problem down wrong.:redface: the pi/2 is outside of the arctan. I need to redo my derivative.
 
  • #8
So here is the corrected derivative. How can I use it to prove the identity?
 

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  • IMAG0032.jpg
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  • #9
Seems like the correct derivative. Now you need to rewrite it until it becomes obvious that it's equal. Note that your equation is certainly equivalent with

[tex]\frac{\sqrt{1-\left(\frac{x-1}{x+1}\right)^2}(x+1)^2}{2}=\sqrt{x}(x+1)[/tex]

Now, maybe try to square both sides?
 
  • #10
I tried simplifying but then this happened.
 

Attachments

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  • #11
The mistake is in your third step. That should read

[tex]\frac{\left(1-\left(\frac{x-1}{x+1}\right)^2\right)(x+1)^2}{4}=x[/tex]

You forgot some brackets...
 
  • #12
ok I simplified both sides to where x = x. Thank You, I can take it from here.:cool:
 

FAQ: Identity Proofs of Inverse Trig Functions

What is an identity proof of an inverse trig function?

An identity proof of an inverse trig function is a mathematical process used to show that two expressions involving inverse trigonometric functions are equivalent. This means that they have the same value for all possible inputs.

Why are identity proofs of inverse trig functions important?

Identity proofs of inverse trig functions are important because they allow us to simplify and manipulate complex trigonometric expressions. They also help us to better understand the relationships between different inverse trig functions.

How do you prove an identity involving inverse trig functions?

To prove an identity involving inverse trig functions, you need to use the properties and identities of trigonometric functions, as well as algebraic manipulation and substitution. You also need to be familiar with the unit circle and the definitions of inverse trig functions.

What are some common strategies for proving inverse trig function identities?

Some common strategies for proving inverse trig function identities include using the Pythagorean identities, converting all trig functions to sine and cosine, and using the double angle and half angle formulas. It is also helpful to break down the expression into smaller parts and work on each part separately.

What are some tips for successfully proving inverse trig function identities?

Some tips for successfully proving inverse trig function identities include: always start with the more complex side of the equation, carefully simplify and manipulate each expression, use the correct identities and formulas, and check your work by substituting in different values for the variables.

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