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