Cho x; y; z khác +- 1 và xy + yz + xz = 1. Chọn câu đúng

* Hướng dẫn giải

$$\begin{gathered} \frac{\text{x}}{\text{1}-{\text{x}}^{\text{2}}}+\frac{\text{y}}{\text{1}-{\text{y}}^{\text{2}}}+\frac{\text{z}}{\text{1}-{\text{z}}^{\text{2}}} \\ =\frac{\text{x(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}+\text{y(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}+\text{z(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ =\frac{\text{x(1}-{\text{z}}^{\text{2}}{\text{-y}}^{\text{2}}+{\text{z}}^{\text{2}}{\text{y}}^{\text{2}}\text{)}+\text{y(1}-{\text{x}}^{\text{2}}-{\text{z}}^{\text{2}}+{\text{x}}^{\text{2}}{\text{z}}^{\text{2}}\text{)}+\text{z(1}-{\text{x}}^{\text{2}}-{\text{y}}^{\text{2}}+{\text{x}}^{\text{2}}{\text{y}}^{\text{2}}\text{)}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ =\frac{\text{x}-{\text{xz}}^{\text{2}}{\text{-xy}}^{\text{2}}+{\text{xy}}^{\text{2}}{\text{z}}^{\text{2}}+\text{y}-{\text{x}}^{\text{2}}\text{y}-{\text{yz}}^{\text{2}}+{\text{yz}}^{\text{2}}{\text{x}}^{\text{2}}+\text{z}-{\text{zx}}^{\text{2}}-{\text{zy}}^{\text{2}}+{\text{zx}}^{\text{2}}{\text{y}}^{\text{2}}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ =\frac{{\text{(x-yx}}^{\text{2}}-{\text{x}}^2\text{z)}+\text{(y}-{\text{xy}}^{\text{2}}-{\text{zy}}^{\text{2}}\text{)}+\text{(z}-{\text{xz}}^{\text{2}}-{\text{yz}}^{\text{2}}\text{)}+{\text{(xy}}^{\text{2}}{\text{z}}^{\text{2}}+{\text{yz}}^{\text{2}}{\text{x}}^{\text{2}}+{\text{zx}}^{\text{2}}{\text{y}}^{\text{2}}\text{)}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ =\frac{\text{x(1-xy}-x\text{z)}+\text{y(1}-\text{xy}-\text{yz)}+\text{z(1}-\text{xz}-\text{zy)}+\text{xyz(yz}+\text{xz}+\text{xy)}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ = \frac{\text{x.yz}+\text{y.xz}+\text{z.xy}+\text{xyz}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \\ =\frac{\text{4xyz}}{\text{(1}-{\text{x}}^{\text{2}}\text{)(1}-{\text{y}}^{\text{2}}\text{)(1}-{\text{z}}^{\text{2}}\text{)}} \end{gathered}$$

Đáp án cần chọn là: C