Question

\(\left(1-\dfrac{5+\sqrt{5}}{1+\sqrt{5}}\right)\left(\dfrac{5-\sqrt{5}}{1-\sqrt{5}}-1\right)\)

Answer 1

`(1 - (5+sqrt{5})/(1+sqrt{5})).((5-sqrt{5})/(1-sqrt{5}) - 1)`

`= (1 - (sqrt{5} . (1+sqrt{5}))/(1+sqrt{5})).((sqrt{5} . (sqrt{5}-1))/(1-sqrt{5}) - 1)`

`=  (1 - sqrt{5}).(-sqrt{5}  - 1)`

`= (sqrt{5}-1).(sqrt{5} + 1)`

`=5 - 1 `

= 4`

Answer 2

\(...=\left(\dfrac{1+\sqrt{5}-5-\sqrt{5}}{1+\sqrt{5}}\right)\left(\dfrac{5-\sqrt{5}-1+\sqrt{5}}{\left(1-\sqrt{5}\right)}\right)\)

\(=\left(\dfrac{-4}{1+\sqrt{5}}\right)\left(\dfrac{4}{\left(1-\sqrt{5}\right)}\right)\)

\(=\dfrac{-16}{1-5}=4\)