+0

# help

0
98
1

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

Dec 19, 2019

#1
+120
0

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