\(\dfrac{x-1}{x+1}-\dfrac{x+1}{x-1}\\ =\dfrac{\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)}-\dfrac{\left(x+1\right)^2}{\left(x+1\right)\left(x-1\right)}\\ =\dfrac{\left(x^2-2x+1\right)}{x^2-1}-\dfrac{\left(x^2+2x+1\right)}{x^2-1}\\ =\dfrac{x^2-2x+1-x^2-2x-1}{x^2-1}\\ =-\dfrac{4x}{x^2-1}\)
\(\dfrac{x-1}{x+1}-\dfrac{x+1}{x-1}=\dfrac{\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)}-\dfrac{\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}=\dfrac{\left(x-1\right)^2-\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}=\dfrac{x^2-2x+1-x^2+2x+1}{\left(x-1\right)\left(x+1\right)}=\dfrac{2}{\left(x-1\right)\left(x+1\right)}\)
