It is given that `cosec theta = -17/15` and `cot beta = -3/4`

`cosec theta = -17/15`

=> `sin theta = -15/17`

As `theta` is in the 4th quadrant `cos theta` is positive.

`cos theta = 8/17`

`cot beta = -3/4`

`1 + cot^2 beta = csc^2 beta`

As `beta` is in the second quadrant `cos beta` is negative and `sin beta` is positive.

`csc beta = 5/4` , `sin beta = 4/5`

`cos beta = -3/5`

`sin(theta + beta) = sin theta*cos beta + cos theta*sin beta`

=> `(-15/17)*( -3/5) + (8/17)*(4/5)`

=>`77/85`

The value of `sin (theta + beta) = 77/85`