Chapter 5, Problem 12CT

To determine

To simplify the given expression of $\frac{1}{s-1}-\frac{2}{s^2-1}$

Answer to Problem 12CT

The answer is $\frac{1}{s+1}$

Explanation of Solution

Given that: : $\frac{1}{s-1}-\frac{2}{s^2-1}$

To expand use $[a^2-b^2]=(a-b)(a+b)$

  $\frac{1}{s-1}-\frac{2}{s^2-1}=\frac{1}{s-1}-\frac{2}{(s-1)(s+1)}$

Multiply and divide both side by $(s+1)$ and simplify

  $\begin{array}{l}=\frac{s+1}{s+1}\left[\frac{1}{s-1}-\frac{2}{(s-1)(s+1)}\right]\\ =\frac{s+1}{s+1}\times \frac{1}{s-1}-\frac{s+1}{s+1}\times \frac{2}{(s-1)(s+1)}\\ =\frac{s+1}{s+1}\times \frac{1}{s-1}-\frac{2}{(s-1)(s+1)}\\ =\frac{(s+1)-2}{(s-1)(s+1)}\\ =\frac{(s-1)}{(s-1)(s+1)}\\ =\frac{1}{(s+1)}\end{array}$

Hence, the simplified answer is $\frac{1}{s+1}$