Just like any other adding and subtracting with unlike denominators, you need to find the least common denominator. In this case, the LCD is just the multiple of the denominators. So multiply both the top and bottom of a/a+1 by (a-1) and multiply both the top and bottom of 1/a-1 by (a+1) to get:

(a/a+1)*(a-1/a-1) - (1/a-1)*(a+1)/(a+1)

= a(a-1)/(a+1)(a-1) - (a+1)/(a+1)(a-1)

Then combine the numerators:

= a(a-1)-(a+1) / (a-1)(a+1)

Distribute and simplify the numerator to get:

a² - a - a - 1 / (a-1)(a+1)

= a² - 2a  - 1 / (a-1)(a+1)

Factor the numerator:

= (a-1)(a-1) / (a-1)(a+1) 

(a-1) in both numerator and denominator cancels out:

= (a-1)/(a+1)

(a - b)(a + b) = a² - b²
~~~~~~~~~~~~
a/(a+1) - 1/(a-1) ; LCD (the least common denominator) is (a + 1)(a - 1) = a² - 1
a(a - 1) / (a² - 1) - (a + 1) / (a² - 1) = (a² - a - a - 1) / (a² - 1) = (a² - 2a -1)/(a² - 1)