Better yet, why don't you show all of YOUR work, i.e. what you have tried and where exactly you are stuck... Then you might get the help you need...
This what I have done:
cscθ = -17/15 where 270° ≤ θ ≤ 360°
sinθ = 1/cscθ
sinθ = 1/(-17/15)
sinθ = 1/1 / (-17/15)
sinθ = 1/1 * (-15/17)
sinθ = -15/17
θ = sin^-1 (-15/17)
** I don't know what to do from here because if I solve for θ completely I won't have an EXACT value **
cot β = - 3/4 where 90° ≤ β ≤ 180°
tanβ = 1/cotβ
tanβ = 1/(-3/4)
tanβ = 1/1 / (-3/4)
tanβ = 1/1 * (-4/3)
tanβ = -4/3
β = tan^-1 (-4/3)
** I don't know what to do from here because if I solve for β completely I won't have an EXACT value **
From this point I'm not sure how to find an exact value for sin(θ + β) because I do not have values for θ or β
I knew about the sum and difference formulas and that I would have to use the sum formula for sin in order to solve the final step, but I'm not sure what you mean by pythagorean identities. I'm not sure how to solve for sin(\theta), cos(\theta), sin(\beta), or cos(\beta).