Simplify the expression (1+sqrt(3))(1-sqrt(3)).

This expression is in the form $(a+b)(a-b)$, which simplifies to $a^2 - b^2$.
Here, $a=1$ and $b=\sqrt{3}$.

$$(1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2$$ $$= 1 - 3$$ $$= -2$$

The simplified expression is $\boxed{-2}$.

To evaluate $\frac{1}{1+\sqrt{3}}$, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $(1-\sqrt{3})$.

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

Using the result from Step 1, we know that $(1+\sqrt{3})(1-\sqrt{3}) = -2$.

$$= \frac{1-\sqrt{3}}{-2}$$ $$= \frac{-( \sqrt{3}-1)}{-2}$$ $$= \frac{\sqrt{3}-1}{2}$$

$$\frac{\sqrt{3}-1}{2} = \frac{1.7321 - 1}{2}$$ $$= \frac{0.7321}{2}$$ $$= 0.36605$$

The first three significant figures are 3, 6, and 6. The fourth digit is 0, so we do not round up.

$$0.36605 \approx 0.366$$

The value of $\frac{1}{1+\sqrt{3}}$ to 3 significant figures is $\boxed{0.366}$.