# Why Does This Algebraic Identity Work in Relativistic Doppler Calculations?

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• Daniel Sellers
In summary, the conversation discusses an algebra identity that has resurfaced while studying the relativistic doppler effect for light. The identity is (1+x)/(sqrt(1-x^2) = sqrt((1+x)/(1-x)) or (sqrt(1-x^2)/(1+x) = sqrt((1-x)/(1+x)). The conversation explains that the identity can be derived from the formula a^2-b^2=(a-b)(a+b) and basic exponent rules. A suggestion is also given to quickly see the validity of the identity by taking the square of both sides. The conversation ends with a humorous comment about easily cancelling out exponents to solve the identity.
Daniel Sellers
I seem to remember this Algebra identity being covered in one of my classes years ago, but it has cropped back up in studying the relativistic doppler effect for light.

Can anyone please show me the intermediate steps to show that:

(1+x)/(sqrt(1-x^2) = sqrt((1+x)/(1-x))

or similarly

(sqrt(1-x^2)/(1+x) = sqrt((1-x)/(1+x))

I can solve problems well enough by factoring gamma out of these equations but it is bugging me that all the texts I can find keep taking this for granted and I can't see why.

## 1-x^2 =(1-x)(1+x) ##. Comes from ## a^2-b^2=(a-b)(a+b) ##. The rest is just things like ## \frac{u^1}{u^{1/2}}=u^{1/2} ## etc. where ## u^{1/2}=\sqrt{u} ##.

Daniel Sellers
I knew it was something obnoxiously simple and obvious! Thanks very much!

Suggestion: to see quickly you can take the square from both side ## \frac{(1+x)^2}{1-x^2}\,=\, \frac{1+x}{1-x}##, now it is quite obvious ...
Ssnow

Ssnow said:
Suggestion: to see quickly you can take the square from both side ## \frac{(1+x)^2}{1-x^2}\,=\, \frac{1+x}{1-x}##, now it is quite obvious ...
Ssnow
Easy -- just cancel the exponents!
$$\frac{(1 + x)^2}{1 - x^2} = \frac{(1 + x)^{\rlap{/}2}}{1 - x^{\rlap{/}2}} = \frac{1 + x}{1 - x}$$

## 1. What is an algebra identity?

An algebra identity is a mathematical equation that is always true, regardless of the values of the variables involved. In other words, it is an equality that holds for all possible values of the variables.

## 2. How do I know if an equation is an algebra identity?

To determine if an equation is an algebra identity, you can substitute different values for the variables and see if the equation remains true. If it does, then it is an identity.

## 3. Can I solve algebra identities?

No, algebra identities cannot be solved because they are already equal. They are used to simplify and manipulate equations, but not to find specific values for the variables.

## 4. What is the purpose of algebra identities?

Algebra identities are used to simplify and manipulate equations in order to solve problems more efficiently. They also help to establish relationships between different mathematical expressions.

## 5. Are there different types of algebra identities?

Yes, there are different types of algebra identities such as the distributive property, commutative property, associative property, and many others. Each type has its own specific form and purpose.

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