If a ( 2 + 3 ) = b ( 2 - 3 ) = 1, then the value of 1a^2 + 1 + 1b^2 + 1 = ?

Answer: Option B

$$\begin{array}{rl}& a\left(2+\sqrt{3}\right)=b\left(2-\sqrt{3}\right)=1\\ & a=\frac{1}{\left(2+\sqrt{3}\right)}\\ & b=\frac{1}{\left(2-\sqrt{3}\right)}\\ & \Rightarrow a=\frac{1}{b}\\ & \Rightarrow \frac{1}{a^2+1}+\frac{1}{b^2+1}\\ & \Rightarrow \frac{1}{\frac{1}{b^2}+1}+\frac{1}{b^2+1}\\ & \Rightarrow \frac{1}{\frac{1+b^2}{b^2}}+\frac{1}{b^2+1}\\ & \Rightarrow \frac{b^2}{b^2+1}+\frac{1}{b^2+1}\\ & \Rightarrow \frac{b^2+1}{b^2+1}\\ & \Rightarrow 1\end{array}$$