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Evaluate 1/(1-x^2) -1 - 1/(x^-2 -1)
alphonsio

We need to evaluate the expression:

11−x2−1−1x−2−1\frac{1}{1 - x^2} - 1 - \frac{1}{x^{-2} - 1}

Step 1: Simplify x−2−1x^{-2} - 1

Since x−2=1x2x^{-2} = \frac{1}{x^2}, we rewrite:

x−2−1=1x2−1=1−x2x2x^{-2} - 1 = \frac{1}{x^2} - 1 = \frac{1 - x^2}{x^2}

Thus, the third term becomes:

1x−2−1=11−x2x2=x21−x2\frac{1}{x^{-2} - 1} = \frac{1}{\frac{1 - x^2}{x^2}} = \frac{x^2}{1 - x^2}

Step 2: Rewrite the Expression

Now, the given expression is:

11−x2−1−x21−x2\frac{1}{1 - x^2} - 1 - \frac{x^2}{1 - x^2}

Step 3: Common Denominator

Rewrite 11 with a denominator of 1−x21 - x^2:

1=1−x21−x21 = \frac{1 - x^2}{1 - x^2}

Now, rewrite the entire expression:

11−x2−1−x21−x2−x21−x2\frac{1}{1 - x^2} - \frac{1 - x^2}{1 - x^2} - \frac{x^2}{1 - x^2}

Step 4: Combine Terms

Since all terms have the same denominator 1−x21 - x^2, combine the numerators:

1−(1−x2)−x21−x2\frac{1 - (1 - x^2) - x^2}{1 - x^2}

Simplify the numerator:

1−1+x2−x2=01 - 1 + x^2 - x^2 = 0

Thus, the entire fraction simplifies to:

01−x2=0\frac{0}{1 - x^2} = 0

Final Answer:

0\boxed{0}