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The simplified form of [((a+1)^2/(a-1)^2...

The simplified form of `[((a+1)^2/(a-1)^2+3)/((a-1)^2/(a+1)^2+3)]/((a^3+1)/(a^3-1)-(2a)/(a-1)]` is:
a.   `a-1`
b.   `1-a`
c.   `-1`
d.   None of these

Text Solution

Verified by Experts

`(((a+1)^2/(a-1)^2+3)/((a-1)^2/(a+1)^2+3))/((a^3+1)/(a^3-1)) - (2a)/(a-1)`
`=((((a+1)^2+3(a-1)^2)/(a-1)^2)/(((a-1)^2+3(a+1)^2)/(a+1)^2))/((a^3+1)/(a^3-1) )- (2a)/(a-1)`
`=(((a^2+1+2a+3a^2+3-6a)/(a-1)^2)*((a+1)^2/(a^2+1-2a+3a^2+3+6a)))/((a^3+1)/(a^3-1)) - (2a)/(a-1)`
`=(((4(a^2-a+1))/(a-1)^2)*((a+1)^2/(4(a^2+a+1))))/((a^3+1)/(a^3-1)) - (2a)/(a-1)`
Now, `(a-1)(a^2+a+1) = a^3 -1 and (a+1)(a^2-a+1) = a^3+1`
So, it becomes,
`=(((a^3+1)(a+1))/((a^3-1)(a-1)))/((a^3+1)/(a^3-1)) - (2a)/(a-1)`
`=(a+1)/(a-1)-(2a)/(a-1)`
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