If-f-x-ln-1-x-1-x-then-prove-that-f-2x-1-x-2-2f-x-

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

If f (x) = \ln (\frac{1 + x}{1 - x}) then prove that

f (\frac{2 x}{1 + x^{2}}) = 2 f (x) .

Commented by mokys last updated on 27/Aug/23

f (\frac{2 x}{1 + x^{2}}) = l n (\frac{1 + \frac{2 x}{1 + x^{2}}}{1 - \frac{2 x}{1 + x^{2}}}) = l n (\frac{1 + 2 x + x^{2}}{1 - 2 x + x^{2}})

= l n {(\frac{1 + x}{1 - x})}^{2} = 2 l n (\frac{1 + x}{1 - x}) = 2 f (x)

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