How is it that (1/(1+(x/a)^2) *(1/a) is equal to a/(a^2+x^2)?

(1/(1 + (x/a)^2) * (1/a)

1/(1 + x^2/a^2) * (1/a)

1/(a + x^2/a) Now multiply by a/a.

a/(a^2+x^2)

A big part of math is pattern recognition. A lot of homework is just fighting with stuff so you will remember the pattern when you see it again. The most common pattern is (x + a) * (x + b) = x^2 + (a + b)x + ab and the special case (x + a) * (x - a) = x^2 - a^2. When you spot the pattern you can just write the answer from memory. The a/a trick is how you rationalize a fraction with an irrational denominator. 1/√2 = √2/2