\(=\dfrac{\left(x-1\right)\left(x+1\right)}{2\left(x-1\right)^2}-\dfrac{x+3}{2\left(x+1\right)}\)
\(=\dfrac{x+1}{2\left(x-1\right)}-\dfrac{x+3}{2\left(x+1\right)}\)
\(=\dfrac{x^2+2x+1-x^2-2x+3}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{2}{x^2-1}\)