1.

If tanalpha,tanbeta are the roots of the equation x^(2)+px+q=0(pne0),then

Answer»

`sin^(2)(alpha+beta)+p sin(alpha+beta)cos(alpha+beta)+qcos^(2)(alpha+beta)=q`
`TAN(alpha+beta)=(q)/(p-1)`
`cos(alpha+beta)=1-q`
`sin(alpha+beta)=-p`

Solution :Since `tanalpha,tanbeta` are the ROOTS of the equation `x^(2)+px+q=0`.
`thereforetanalpha+tanbeta=-p,tanalphatanbeta=q`
`thereforetan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)=(p)/(q-1)`
Also ,when `tan(alpha+beta)=(p)/(q-1)`.
L.H.S. of the EXPRESSION given in
`=cos^(2)(alpha+beta)[tan^(2)(alpha+beta)+ptan(alpha+beta)+q]`
`=(1)/(1+tan^(2)(alpha+beta))[(p^(2))/((q-1)^(2))+(p^(2))/(q-1)+q]`
`=((q-1)^(2))/((q-1)^(2)-p^(2))[(p^(2)+p^(2)(q-1)+q(q-1)^(2))/((q-1)^(2))]`
`=(q{p^(2)+(q-1)^(1)})/(p^(2)+(q-1)^(2))`
=q = R.H.S.of
i.e., Relation given in (a) is also SATISFIED.


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