We have to prove that (x/(x+1)+1):(1-3x^2/(1-x^2))=(1-x)/(1-2x)

Starting with the left hand side

(x/(x+1)+1):(1-3x^2/(1-x^2))

=> [(x/(x+1)+1)]/[(1-3x^2/(1-x^2))]

=> [(x+x+1)/(x+1)]/[(1-x^2-3x^2)/(1-x^2)]

=> [(2x+1)/(x+1)]/[(1-4x^2)/(1-x^2)]

=> [(2x+1)/(x+1)]/[(1-2x)(1+2x)/(1-x)(1+x)]

=> [(2x+1)(1-x)(1+x)/(x+1)(1-2x)(1+2x)]

=> [(1-x)/(1-2x)]

which is the right hand side

**This proves that (x/(x+1)+1):(1-3x^2/(1-x^2))=(1-x)/(1-2x)**

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We have to prove that (x/(x+1)+1):(1-3x^2/(1-x^2))=(1-x)/(1-2x)

Starting with the left hand side

(x/(x+1)+1):(1-3x^2/(1-x^2))

=> [(x/(x+1)+1)]/[(1-3x^2/(1-x^2))]

=> [(x+x+1)/(x+1)]/[(1-x^2-3x^2)/(1-x^2)]

=> [(2x+1)/(x+1)]/[(1-4x^2)/(1-x^2)]

=> [(2x+1)/(x+1)]/[(1-2x)(1+2x)/(1-x)(1+x)]

=> [(2x+1)(1-x)(1+x)/(x+1)(1-2x)(1+2x)]

=> [(1-x)/(1-2x)]

which is the right hand side

**This proves that (x/(x+1)+1):(1-3x^2/(1-x^2))=(1-x)/(1-2x)**