Now compute $ x^2 + 1 $ and $ x^2 - 1 $:
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- y + \frac{1}{y}
- We compute $ y + \frac{1}{y} $. First, recall that $ x + \frac{1}{x} = 5 $. Square both sides:
- x^2 + 2 + \frac{1}{x^2} = 25 \Rightarrow x^2 + \frac{1}{x^2} = 23
- Let $ A = x^2 + 1 $, $ B = x^2 - 1 $. Then:
- \frac{A}{B} + \frac{B}{A} = \frac{(x^2 + 1)^2 + (x^2 - 1)^2}{(x^2 - 1)(x^2 + 1)}
- Compute numerator:
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