If a (2+ √3) = b (2 - √3) then the value of 1/(a^2+1)+1/(b^2+1) is ?->(Show Answer!)

1. If a (2+ √3) = b (2 - √3) then the value of 1/(a^2+1)+1/(b^2+1) is

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By: anil on 05 May 2019 01.53 am

Let $$a =(2-sqrt{3})$$ and $$b =(2 + sqrt{3})$$
$$a^{2}= 7- 4sqrt{3}$$, and $$b^{2}= 7+4sqrt{3}$$ $$ frac{1}{a^2+1}+frac{1}{b^2+1} = frac{(a)^{2} + (b)^{2} +2}{((a)^{2}+1)((b)^{2}+1)}$$ $$(a^{2}+1)(b^{2}+1)$$ = $$(8-4sqrt{3})(8+4sqrt{3})$$ = $$16(2-sqrt{3})(2+sqrt{3}) = 16$$

$$a^{2} + b^{2} +2= 16$$
Thus $$ frac{1}{a^{2}+1}+frac{1}{b^{2}+1}=1$$

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