Question

If \( A = \frac{x+1}{x-1} \) and \( B = \frac{x-1}{x+1} \), then \( A + B \) is:

• A. None of these
• B. \( \frac{2(x^2 + 1)}{(x - 1)^2} \)
• C. \( \frac{2(x^2 - 1)}{x^2 + 1} \)
• D. \( \frac{x^2 + 1}{x^2 - 1} \)

Hint:

When adding fractions, always find a common denominator and simplify the expression carefully.

Answer 1

The Correct Option is B

Given expressions for \( A \) and \( B \), we first add them: \[\nA + B = \frac{x+1}{x-1} + \frac{x-1}{x+1}.\n\] The common denominator is \( (x-1)(x+1) = x^2 - 1 \). Therefore: \[\nA + B = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)}.\n\] Expanding the numerator: \[\n(x+1)^2 + (x-1)^2 = x^2 + 2x + 1 + x^2 - 2x + 1 = 2x^2 + 2.\n\] Hence: \[\nA + B = \frac{2(x^2 + 1)}{x^2 - 1}.\n\] The answer is \( \frac{2(x^2 + 1)}{x^2 - 1} \), corresponding to option (B).