<span>So we need to find the fraction of many times does purple show up in 20 trays in which every tray has 5 colors. To find the answer we need to find the overall number of all colors and that is 20 trays * 5 colors and that equals to 100 colors. So purple shows 20 times and 20/100 is 1/5 when we simplify the fraction. </span>
This is one pathway to prove the identity.
Part 1

Part 2

Part 3

As the steps above show, the goal is to get both sides be the same identical expression. You should only work with one side to transform it into the other. In this case, the left side transforms while the right side stays fixed the entire time. The general rule is that you should convert the more complicated expression into a simpler form.
We use other previously established or proven trig identities to work through the steps. For example, I used the pythagorean identity
in the second to last step. I broke the steps into three parts to hopefully make it more manageable.