In a projectile motion let (x,y) be any point , it makes alpha and beta angles with horizontal plane - Physics - Motion In A Plane

Question

In a projectile motion let (x,y) be any point , it makes alpha and
beta angles with horizontal plane Prove that tan theta= tan
alpha+tan beta where as theta is the angle of projection
HINT {HORIZONTAL RANGE CAN BE DIVIDED INTO X AND R-X AND TAN
ALPHA=Y÷X AND TAN BETA=Y÷(R-X)}

the question may not be clear to you but plzz try to prove it

Ved Prakash Lakhera · Accepted Answer

Let us consider a projectile is projected with velocity u making an angle θ with the horizontal
At any instant of time  let the position of projectile be x,y such that this position makes angle αand β with the horizontal plane as shown in the diagram
 Therefore,h =y = x tanα = ( R -x ) tanβor    yx  = tanα             and        yR-x = tanβ
adding these two equations we get ,  y1x+1R-x = tanα + tanβ
y Rx R-x = tanα + tanβ
We now  R = 2u2gsinθ cosθ and     y = x tanθ -gx22u2cos2θ
 So,y Rx R-x =  x tanθ -gx22u2cos2θ×2u2gsinθ cosθx2u2gsinθ cosθ -x=(2u2sinθ cosθ-gx) sinθ×1cosθ(2u2sinθ cosθ-gx) =tanθ
Therefore    tanθ = tanα + tanβ