Solve `sinx+siny=sin(x+y)a n d|x|+|y|=1`

Explanation

`sin x + sin y= sin (x+y)`
or `2 "sin" (x+y)/2 ["cos" (x-y)/2-"cos" (x+y)/2]=0`
or `4"sin" (x+y)/2 "sin" x/2"sin" y/2=0`
(a) `"sin" (x+y)/2=0`
`rArr x+y=2npi, n in Z`
`rArr x+y=0" "( :' |x|+|y|=1 rArr -1 le x, y le 1)`
(b) `"sin" x/2=0`
`rArr x=2 mpi, m in Z`
`rArr x=0`
(c) `"sin" y/2 =0`
`rArr y=2p pi, p in Z`
`rArr y=0`
From `|x|+|y|=1`
If `x=0`, then `|y|=1`
`rArr y= pm 1`
If `y=0`, then `|x|=1`
`rArr x= pm 1`
If `y=-x`, then `|x|+|-x|=2`
`rArr x= pm 1/2 and y= pm 1/2`
Hence, solutions are
`(0, 1), (0, -1), (1, 0), (-1, 0), (1/2, -1/2)`, and `(-1/2, 1/2)`.