Solution
Given, $$A=\frac{x-1}{x+1}$$
$$A-\frac{1}{A}$$ = $$\frac{x-1}{x+1}-\frac{x+1}{x-1}$$
$$=\frac{\left(x-1\right)^2-\left(x+1\right)^2}{x^2-1}$$
$$=\frac{x^2-2x+1-\left(x^2+2x+1\right)^{ }}{x^2-1}$$
$$=\frac{x^2-2x+1-x^2-2x-1}{x^2-1}$$
$$=\frac{-4x}{x^2-1}$$
Hence, the correct answer is Option D
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