if tanA and tanB are the roots of x^2+px+q=0, show that the value of sin^2(A+B)+p.sin(A+B).cos(A+B)+q.cos^2(A+B) =q.

Equate p and q with the expressions of tan A and tan B using the sum and product of roots of a quadratic eqn.. Substitute them in the last eqn to obtain the equality. Yuo can start by dividing the whole eqn by Sin²(A+B). And calculate tan (A+B) in terms of p and q.