Sinx+cosx= - sqaure root of 2

the end answer should be x=-135+360k

Squaring both sides:-

sin^2x + cos^2 x + 2 sinx cosx  = 2
1 + 2 sinx cos x = 2

2 sinx cos x =  1

sin2x = 1

2x =  90 degrees

so x = 45 degrees

but this is not the root  because   the original equation has a negative right side

sin45 + cos 45 =  + sqrt2

the only quadrant in which the sine and cosine is negative is the third quadrant

so x =  -180 + 45 = -135

general case  x = -136 + 36k