Question

If α and β are distinct roots of the equation pcos x +qsin x =r, then show that tan (α+β)/2 = q/p

Answer 1

a and ß are the roots of

Pcosx + qsinx = r
so,

Pcosa + qsina = r -------;(1)

Pcosß + qsinß = r --------(2)

subtract eqns (1) -(2)

P{ cosa - cosß } + q { sina - sinß } =0

P { cosa - cosß } = q { sinß - sina}

P{ 2sin(a + ß)/2 sin(ß -a)/2 } = q { 2sin(ß -a)/2 .cos( a + ß)/2}

P.sin( a + ß)/2 = q cos(a + ß)/2

tan( a + ß)/2 = q/P