Đáp án: 
Ta có:
$\sqrt{1+\frac{1}{x^2}+\frac{1}{(x+1)^2}}$
$=\sqrt{\frac{x^2(x+1)^2+(x+1)^2+x^2}{x^2(x+1)^2}}$
$=\sqrt{\frac{(x(x+1))^2+x^2+2x+1+x^2}{x^2(x+1)^2}}$ 
$=\sqrt{\frac{(x(x+1)^2+2x(x+1)+1}{x^2(x+1)^2}}$
$=\sqrt{\frac{(x(x+1)+1)^2}{x^2(x+1)^2}}$
$=\frac{x(x+1)+1}{x(x+1)}$
$=1+\frac{1}{x(x+1)}$
$=1+\frac{1}{x}-\frac{1}{x+1}$ (đpcm)