1. We know that sin(x)=-3/5, cos(y)=-7/25, and that x and y are in quadrant 3

2. We know that sin(x) and cos(y) are negative (this is given), so cos(x) and sin(y) will also be negative, because both x and y are in the third quadrant. Since sin(x) is -3/5, cos(x) will be -4/5, because sin²(x) + cos²(x) must be =1.  Same applies to cos(y) and sin(y)…so sin(y) will be -24/25. (We skipped a few steps here, so let us know if you want us to break down the steps in more detail.)

3. Using these inputs we will have 

sin(x+y)= sin(x) cos(y) +cos(x) sin(y)

           = (-3/5)(-7/25) + (-4/5)(-24/25)

          = 21/125+96/125  =117/125

cos(x-y)= cos(x)cos(y)+sin(x)sin(y)

           = (-4/5)(-7/25)+(-3/5)(-24/25)

          =28/125+72/125= 100/125=4/5

4. In order to calculate tan(x+y) , cos(x+y) is also needed, which can be computed using formula cos(x+y)= cos(x)cos(y)-sin(x)sin(y).

5. Hence cos(x+y)= 28/125-72/125 = -44/125.

6. Now tan(x+y)=sin(x+y)/cos(x+y)= 117/125/ (-44)/125  = -117/44

7. Quadrant in which x+y will lie can be determined by the fact that sin(x+y) is positive and cos(x+y) is negative. This situation exists in the 2nd Quadrant, hence we can say that x+y lies in the second quadrant.