a particle is thrown over a triangle from one end of a horizontal base and gauging the vertex falls on other end of the base .if a and b are base angles and c the angle of projection, prove that
tan c = tan a + tan b

I think you mean "grazing the vertex," not "gauging" - right?

This is essentially a math probem -- the ony physics here is in recognizing that the path taken by the thrown particle is a parabola. Let's call the length of the base B, and the maximum height of the parabola H. If we center the parabola on the y-axis, its equation is:


y-h = -\frac {4H} {B^2} x^2


The tangent of angle "a" is \frac y {(B/2+x)} and the tangent of angle "b" is \frac y {(B/2-x)}. You want to show that the sum of tan(a) and tan(b) is the derivative of the parabola at x = -b/2, which is equal to tan(c). Can you take it from here?