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Prove this identity

  1. Jan 10, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    If [itex]sin^{-1}x+sin^{-1}y+sin^{-1}z = \pi [/itex] then prove that [itex]x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz [/itex]

    2. Relevant equations

    3. The attempt at a solution

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

    [itex]tan(\theta + \alpha) = tan(\pi - \beta) \\

    tan(\theta + \alpha) = -tan(\beta)
    [/itex]

    After simplifying I get something like this
    [itex]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[/itex]

    I know it's close but it is not yet the final result.
     
  2. jcsd
  3. Jan 10, 2013 #2
    Hello again, utkarshakash! Starting from your second step...

    [itex]\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}}[/itex]

    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 :biggrin:.

    [itex]\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}}[/itex]

    Got it from here?
     
  4. Jan 10, 2013 #3

    haruspex

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    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.
     
  5. Jan 10, 2013 #4

    utkarshakash

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    I am still not getting it. What I have to do after the last step?
     
  6. Jan 10, 2013 #5
    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: Jan 10, 2013
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