一个条件代数等式

青青子衿

\begin{align*}
\frac{1-\frac{X+Y}{1+XY}}{1+\frac{X+Y}{1+XY}}=\left(\frac{1-\frac{X\sqrt{1-Y^2}+Y\sqrt{1-X^2}}{\sqrt{1-X^2}+\sqrt{1-Y^2}}}{1+\frac{X\sqrt{1-Y^2}+Y\sqrt{1-X^2}}{\sqrt{1-X^2}+\sqrt{1-Y^2}}}\right)^2
\end{align*}

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回复 1# 青青子衿
令$X=\tanh x,Y=\tanh Y$,左边化为$e^{2(x+y)}$,
\[\frac{X \sqrt{1-Y^2}+Y\sqrt{1-X^2}}{\sqrt{1-X^2}+\sqrt{1-Y^2}}=\frac{\tanh x \operatorname{sech}y+ \tanh y\operatorname{sech}x}{\operatorname{sech}x+\operatorname{sech}y}=\tanh \left(\frac{x+y}{2}\right)\]右边化为$(e^{x+y})^2$