What is the general method for easily solving these type of expressions? $(a+b)/(a-b)$

Question

$\dfrac{a+b}{a-b}$. the particular expression in my case is:

$$ \frac{20 \sin(x)-1}{20 \sin(x)+1} $$

the value of $\sin x$ is $\dfrac{100}{\sqrt{1000^2-100^2}}$ so it's very hard to do it by simple arithmetic.

Answer 1

Generally we have that $$\frac{a-b}{a+b}=\frac{a+b}{a-b}\frac{a-b}{a-b}=\frac{a^2-b^2}{(a+b)^2}$$ and that $$\frac{a+b}{a-b}=\left(\frac{a-b}{a+b}\right)^{-1}=\frac{(a+b)^2}{a^2-b^2}$$