Question

Simplify $\left(\frac{1+\sqrt{3}}{1-\sqrt{3}}\right)+\left(\frac{1-\sqrt{3}}{1+\sqrt{3}}\right)$

• A. $1$
• B. $-4$
• C. $-2$
• D. $3$

Answer 1

The correct option is B

$-4$

It can be solved by rationalizing the denominators individually. For that, first find the rationalizing factors for both the denominators, ($1-\sqrt{3}$) and ($1+\sqrt{3}$). The rationalizing factor for ($1-\sqrt{3}$) would be ($1+\sqrt{3}$) and that for ($1+\sqrt{3}$) would be ($1-\sqrt{3}$)

Rationalizing the given terms we get,

$\frac{(1+\sqrt{3})(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})}+\frac{(1-\sqrt{3})(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}$

$=\frac{1+3+2\sqrt{3}}{1-3}+\frac{1+3-2\sqrt{3}}{1-3}$

$=\frac{4+2\sqrt{3}}{-2}+\frac{4-2\sqrt{3}}{-2}$

$=-2-\sqrt{3}-2+\sqrt{3}$

$=-4$