Answer :
Step-by-step explanation:
\[ 1 - \left\{ 1 + \left( (x^2 - 1)^{-1} \right)^{-1} \right\}^{-1} \]
First, simplify the innermost expression:
\[ (x^2 - 1)^{-1} = \frac{1}{x^2 - 1} \]
\(\frac{1}{x^2 - 1}\):
\[ \left( \frac{1}{x^2 - 1} \right)^{-1} = x^2 - 1 \]
expression:
\[ 1 + (x^2 - 1) \]
Now, take the reciprocal of \(1 + (x^2 - 1)\):
\[ \left\{ 1 + (x^2 - 1) \right\}^{-1} = \frac{1}{1 + (x^2 - 1)} \]
Substitute this into the outer expression:
\[ 1 - \left( \frac{1}{1 + (x^2 - 1)} \right) \]
To simplify further, simplify the fraction:
\[ 1 - \frac{1}{1 + x^2 - 1} \]
\[ 1 - \frac{1}{x^2} \]
Therefore, the simplified expression is:
\[ \boxed{\frac{x^2 - 1}{x^2}} \]