Question

$1 - \{1 + (x^2 - 1)^{-1}\}^{-1} = ?$}

• A. $x$
• B. $\frac{1}{x}$
• C. $\frac{1}{x^2}$
• D. $x^2$

Hint:

Negative powers invert the fraction—apply carefully step by step.

Answer 1

The Correct Option is C

Concept: Simplify step by step using exponent rules.
Step 1: Simplify inner term.
\[ (x^2 - 1)^{-1} = \frac{1}{x^2 - 1} \]
Step 2: Add inside bracket.
\[ 1 + \frac{1}{x^2 - 1} = \frac{x^2 - 1 + 1}{x^2 - 1} = \frac{x^2}{x^2 - 1} \]
Step 3: Apply power $-1$.
\[ \left(\frac{x^2}{x^2 - 1}\right)^{-1} = \frac{x^2 - 1}{x^2} \]
Step 4: Final simplification.
\[ 1 - \frac{x^2 - 1}{x^2} = \frac{x^2 - (x^2 - 1)}{x^2} = \frac{1}{x^2} \]
Hence, the value is $\frac{1{x^2}$.