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On simplification the value of $${\text{1}} - $$ $$\frac{1}{{1 + \sqrt 2 }}{\text{ + }}$$  $$\frac{1}{{1 - \sqrt 2 }}$$  is = ?

On simplification the value of $${\text{1}} - $$ $$\frac{1}{{1 + \sqrt 2 }}{\text{ + }}$$  $$\frac{1}{{1 - \sqrt 2 }}$$  is = ? Correct Answer $$1 - 2\sqrt 2 $$

  $${\text{1}} - \frac{1}{{1 + \sqrt 2 }}{\text{ + }}\frac{1}{{1 - \sqrt 2 }}{\text{ }}$$
  $$ = 1 - \frac{{\left( {\sqrt 2 - 1} \right)}}{{\left( {\sqrt 2 + 1} \right)\left( {\sqrt 2 - 1} \right)}} - $$     $$\frac{{\left( {\sqrt 2 + 1} \right)}}{{\left( {\sqrt 2 - 1} \right)\left( {\sqrt 2 + 1} \right)}}$$
$$\eqalign{ & = 1 - \sqrt 2 + 1 - \sqrt 2 - 1 \cr & = 1 - 2\sqrt 2 \cr} $$

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