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Find the number of values that satisfy tan(arctan(x) + arctan(x^2)) = x.

 Dec 19, 2019
 #1
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Let

\(\alpha= arctan(x)\), and

 

\(\beta=arctan(x^2)\). Then,

 

\(tan(\alpha)=x\), and \(tan(\beta)=x^2\), and

 

\(tan(arctan(x)+arctan(x^2))=Tan(\alpha+\beta)\)

 

\(=\frac{tan(\alpha)+tan(\beta)}{1-tan(\alpha)tan(\beta)} =\frac{x+x^2}{1-x\cdot x^2}=\frac{x+x^2}{1-x^3}=x\);

 

Cross-multiplying , we get

 

\(x+x^2=x-x^4\), which implies \(x^2=-x^4\). This equation is only true for \(x=0\) (unless we are in the complex field). So there is only one value satisfying the original equation.

 Dec 20, 2019

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