Question

General idea of how to do this

Answer 1

The general approach to this is to write the (x, y) coordinates of the trajectory's peak in terms of the sine and cosine of the launch angle α. Then use a trig identity (sin(2α)² + cos(2α)² = 1) to eliminate the dependence on α.

You should get
   t = v₀sin(α)/g
   x = v₀²sin(2α)/(2g)
   y = v₀²(1-cos(2α))/(4g)

Solve each of the latter two equations for the trig function, then substitute those expressions into the trig identity above. Divide by the coefficient of x² and rearrange to get the expression shown.