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.