solve the system

sinx+cosy=1

cos2x-cos2y=1

cos2x = 1-2sin²x                ......1

cos2y = 2cos²y-1              .........2

we have , cos2x-cos2y = 1

so , after putting  1 & 2 in above eq we get

 2(sin²x + cos²y) = 1

  sin²x+cos²y = 1/2            ...............3

 sinx+cosy = 1        .......................4         (given)

now ,

           (sinx+cosy)² = sin²x + cos²x + 2sinxcosy

                   1²  =  1/2 + 2sinxcosy

                  sinxcosy = 1/4       ................5

now from 4 & 5

                  sinx(1-sinx) = 1/4

                   (2sinx-1)² = 0

        sinx= 1/2                                         cosy = 1/2

         x = npi +(-1)ⁿ (pi/6)                           y = 2npi +(-) (pi/3)

sinx=.5

cosy=.5

 use cos2x=2cos squarex -1 and eliminate cos x.