If sin = 2/3, the side opposite from the angle is 2 and the hypotenuse is 3, since sin is opposite over hypotenuse.
a^2 + b^2 = c^2. The hypotenuse, 3, is c, and 2 can be a. 3^2 - 2^2 is 5. Since that leaves b^2 = 5, you have to take the root of both sides, so b = root5. That means the adjacent side is root5.
Cos is adjacent over hypotenuse, so cos is root5 over 3. Sec is cos flipped, so sec is 3/root5 or 3root5/5. When you square that whole thing, you get 9 times 5 (45) divided by 25, which is the same thing as 9/5.
I hope that helped!
sin^2 = 4/9
cos^2 = 1 - sin^2 = 5/9
1/cos^2 = sec^2 = 9/5