Find the value
If \[ \sqrt{2} = 1.4142,\ \text{then} \quad \sqrt{\frac{\sqrt{2} – 1}{\sqrt{2} + 1}} \]
(a) 0.1718 \quad (b) 5.8282 \quad (c) 0.4142 \quad (d) 2.4142
Solution:
\[ \frac{\sqrt{2} – 1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} = \frac{(\sqrt{2} – 1)^2}{2 – 1} \]
\[ = (\sqrt{2} – 1)^2 \]
\[ \sqrt{\frac{\sqrt{2} – 1}{\sqrt{2} + 1}} = \sqrt{(\sqrt{2} – 1)^2} = \sqrt{2} – 1 \]
\[ = 1.4142 – 1 = 0.4142 \]
\[ \therefore \text{Correct Answer: } 0.4142 \ (\text{Option c}) \]