If x = 3 2 , then the value of ( 1 + x + 1 - x 1 + x - 1 - x )\, is = ?

Answer: Option D

$$\begin{array}{rl}& x=\frac{\sqrt{3}}{2}\\ & =\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\times \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\\ & =\frac{{\left(\sqrt{1+x}+\sqrt{1-x}\right)}^2}{{\left(\sqrt{1+x}\right)}^2-{\left(\sqrt{1-x}\right)}^2}\\ & =\frac{1+x+1-x+2\sqrt{1-x^2}}{1+x-1+x}\\ & =\frac{2+2\sqrt{1-x^2}}{2x}\\ & =\frac{1+\sqrt{1-x^2}}{x}\\ & =\frac{1+\sqrt{1-\frac{3}{4}}}{\sqrt{3}}\times 2\\ & =\frac{\left(1+\frac{1}{2}\right)}{\sqrt{3}}\times 2\\ & =\frac{\frac{3}{2}\times 2}{\sqrt{3}}\\ & =\sqrt{3}\end{array}$$