If x + 1 /x = a +b and x - 1/x = a - b then prove ab =1

Dear Student,

Please find below the solution to the asked query :

$$\begin{gathered} \mathrm{x} + \frac{1}{\mathrm{x}} = \mathrm{a} + \mathrm{b} ....\left(1\right) \\ \mathrm{x} - \frac{1}{\mathrm{x}} = \mathrm{a} - \mathrm{b} ....\left(2\right) \\ \mathrm{Adding} \left(1\right) \mathrm{and} \left(2\right), \mathrm{we} \mathrm{get} \\ 2\mathrm{x} = 2\mathrm{a} \\ \Rightarrow \mathrm{a} = \mathrm{x} \\ \mathrm{Now}, \mathrm{put} \mathrm{a} = \mathrm{x} \mathrm{in} \left(1\right), \mathrm{we} \mathrm{get} \\ \mathrm{x} + \frac{1}{\mathrm{x}} = \mathrm{x} + \mathrm{b} \\ \Rightarrow \mathrm{b} = \frac{1}{\mathrm{x}} \\ \mathrm{Now}, \mathrm{ab} = \mathrm{x} \times \frac{1}{\mathrm{x}} = 1 \end{gathered}$$

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