# Prove: x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz

• utkarshakash
In summary: So, if you're asking how I help people with homework problems that they aren't able to figure out on their own, I would say that I ask them questions on the problem and see if I can figure out what they are trying to do.
utkarshakash
Gold Member

## Homework Statement

If $sin^{-1}x+sin^{-1}y+sin^{-1}z = \pi$ then prove that $x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz$

## The Attempt at a Solution

I assume the inverse functions to be θ, α, β respectively. Rearranging and taking tan of both sides

$tan(\theta + \alpha) = tan(\pi - \beta) \\ tan(\theta + \alpha) = -tan(\beta)$

After simplifying I get something like this
$x\sqrt{(1-y^2)(1-z^2)}+y\sqrt{(1-x^2)(1-z^2)}+z\sqrt{(1-x^2)(1-y^2)} = xyz$

I know it's close but it is not yet the final result.

utkarshakash said:

## Homework Statement

If $sin^{-1}x+sin^{-1}y+sin^{-1}z = \pi$ then prove that $x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz$

## The Attempt at a Solution

I assume the inverse functions to be θ, α, β respectively. Rearranging and taking tan of both sides

$tan(\theta + \alpha) = tan(\pi - \beta) \\ tan(\theta + \alpha) = -tan(\beta)$

After simplifying I get something like this
$x\sqrt{(1-y^2)(1-z^2)}+y\sqrt{(1-x^2)(1-z^2)}+z\sqrt{(1-x^2)(1-y^2)} = xyz$

I know it's close but it is not yet the final result.
Hello again, utkarshakash! Starting from your second step...

$\frac{tan\theta + tan\alpha}{1-tan\alpha tan\theta} = -tan\beta \\ tan\theta + tan\alpha = -tan\beta + tan\alpha \ tan\theta \ tan\beta \\ tan\theta + tan\alpha + tan\beta = tan\alpha \ tan\theta \ tan\beta \\ tan(arcsinx) + tan(arcsiny) + tan(arcsinz) = tan(arcsinx) \ tan(arcsiny) \ tan(arcsinz) \\ \frac{x}{\sqrt{1-x^2}} + \frac{y}{\sqrt{1-y^2}} + \frac{z}{\sqrt{1-z^2}} = \frac{xyz}{\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}}$

Here, I think, is where you went wrong. You want each term on the left hand side to be of the form "n√(1-n^2)". Thus, we multiply each term by 1 .

$\frac{x\sqrt{1-x^2}}{1-x^2} + \frac{y\sqrt{1-y^2}}{1-y^2} + \frac{z\sqrt{1-z^2}}{1-z^2} = \frac{xyz}{\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}}$

Got it from here?

There's probably a more elegant way, but if you simply eliminate z from each side (= x√(1-y2) + y√(1-x2)) then it should become reasonably evident.

Mandelbroth said:
Hello again, utkarshakash! Starting from your second step...

$\frac{tan\theta + tan\alpha}{1-tan\alpha tan\theta} = -tan\beta \\ tan\theta + tan\alpha = -tan\beta + tan\alpha \ tan\theta \ tan\beta \\ tan\theta + tan\alpha + tan\beta = tan\alpha \ tan\theta \ tan\beta \\ tan(arcsinx) + tan(arcsiny) + tan(arcsinz) = tan(arcsinx) \ tan(arcsiny) \ tan(arcsinz) \\ \frac{x}{\sqrt{1-x^2}} + \frac{y}{\sqrt{1-y^2}} + \frac{z}{\sqrt{1-z^2}} = \frac{xyz}{\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}}$

Here, I think, is where you went wrong. You want each term on the left hand side to be of the form "n√(1-n^2)". Thus, we multiply each term by 1 .

$\frac{x\sqrt{1-x^2}}{1-x^2} + \frac{y\sqrt{1-y^2}}{1-y^2} + \frac{z\sqrt{1-z^2}}{1-z^2} = \frac{xyz}{\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}}$

Got it from here?

I am still not getting it. What I have to do after the last step?

I'm sorry for somewhat hijacking this thread, but how do you guys (the homework helpers and the like) help people with problems such as this on a whim? I just got done with Calc1 and I wouldn't even know where to begin with this proof really. For example, I completely forgot that tan(a+b) = tan(a)+tan(b) / (1 - tan(a)tan(b)). How do you guys keep these identities fresh in your mind? Are you teachers or mathematics degree students?

I don't mention the OP because it's different learning something and applying it directly in a problem that you know involves applying what you have recently learned; I'm talking about learning something and being able to retain it long after you have learned it.

Last edited:

## 1. What does this equation mean?

This equation is a mathematical expression that relates three variables, x, y, and z, and their corresponding square roots. It states that when you plug in values for x, y, and z that satisfy the equation, the sum of their square roots will equal 2 times the product of x, y, and z.

## 2. How do I prove this equation?

The equation can be proven using algebraic manipulation and substitution. You can start by simplifying the square roots using the properties of exponents and then rearranging the terms to get the desired result.

## 3. What is the significance of this equation?

This equation is significant because it shows a relationship between three variables and their square roots. It is also a demonstration of how mathematical equations can be proven using logical steps and mathematical operations.

## 4. What is the domain and range of this equation?

The domain of this equation is all real numbers, as long as the values of x, y, and z satisfy the equation. The range is also all real numbers, as the equation can produce any real number as its solution.

## 5. Can this equation be extended to more variables?

Yes, this equation can be extended to include more variables, as long as the number of variables is equal to the number of square roots on the left side of the equation. The equation will still hold true as long as the values of the variables satisfy the equation.

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