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A particle is thrown over a triangle from one end of a horizontal base and after grazing the vertex falls on the other end of the base. If `alpha and beta` be the base angles and `theta` the angle of projection, prove that `tan theta = tan alpha + tan beta` .

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The situation is shown in figure. Form figure, we have

`tan alpha+tan beta=(y)/(x)+(y)/(R-x)`
`tan alpha+tan beta=(yR)/(x(R-x))` ………….(1)
But equation of trajectory is `y=x tan theta[1-(x)/(R )]`
`tan theta=[(yR)/(x(R-x))]`........(ii)
From eqs. (i) and (ii), `tan theta=tan alpha+tan beta`
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