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- Feb 14, 2012

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I've encountered a problem in deciding the condition in order for the equality to hold.

Here is the problem:

If $x\sqrt {1-y^2} + y \sqrt {1-x^2}=1$, prove that $x^2+y^2=1$

By using the Cauchy-Schwarz inequality, it's fairly easy to prove that $x\sqrt {1-y^2} + y \sqrt {1-x^2}\leq1$

Next, what I tried to do is to work backwards and let $x^2+y^2=1$, then I see that $x=\sqrt {1-y^2}$. After making that substitution into the LHS of the inequality $ x\sqrt {1-y^2} + y \sqrt {1-x^2} $ and I eventually get 1 as the final answer.

What do you think, Sir? I feel bad for doing this.

Do you have any idea to deduce the condition from $x\sqrt {1-y^2} + y \sqrt {1-x^2}\leq1$?

Thanks.