How do you find the missing side length for a 45-45-90 triangle?

It kind of depends on which side is missing.
Here are some facts about a 45-45-90 triangle
that will be helpful if you know which one to use:

-- The two "legs" have the same length.

-- Each leg is (1/2 · √2) times (the hypotenuse).

-- The hypotenuse is  (√2) times the length of either leg.

-- For either of the acute angles ...
           ... The tangent of the angle is  1.
           ... The sine and the cosine of the angle are both  (1/2 · √2).