Question

find all the possible triplets `(a_(1), a_(2), a_(3))` such that `a_(1)+a_(2) cos (2x)+a_(3) sin^(2) (x)=0` for all real x.

Answer 1

We have
`a_(1)+a_(2) cos (2x) +a_(3) sin^(2) x=0` for all real x
`:. a_(1)+a_(2) (1-2 sin^(2) x)+a_(3) sin^(2) x=0`
`rArr (a_(1)+a_(2))+(-2a_(2)+a_(3)) sin^(2) x=0`
Since this is true for all real x, we must have
`a_(!)+a_(2)=0` and `-2a_(2)+a_(3)=0`
`:. a_(2)=-a_(1)` and `a_(3)= -2a_(1)`
Thus, there exists infinite triplets `(a_(1), -a_(1), -2a_(1))`