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Question Number 196596 by MATHEMATICSAM last updated on 27/Aug/23

$$\mathrm{If}f\left(x\right)=\ln \left(\frac{1+x}{1-x}\right)\mathrm{then}\mathrm{prove}\mathrm{that}$$$$f\left(\frac{2x}{1+x^2}\right)=2f\left(x\right).$$

Commented by mokys last updated on 27/Aug/23

$$f\left(\frac{2x}{1+x^2}\right)=ln\left(\frac{1+\frac{2x}{1+x^2}}{1-\frac{2x}{1+x^2}}\right)=ln\left(\frac{1+2x+x^2}{1-2x+x^2}\right)$$$$$$$$=ln{\left(\frac{1+x}{1-x}\right)}^2=2ln\left(\frac{1+x}{1-x}\right)=2f\left(x\right)$$

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