solve the equation
cos(x) + sin(x) = 1
square both sides
cos^2 x + 2sinxcosx + sin^2 x = 1
1 + 2sinxcosx = 1
sin (2x) = 0
2x = 0 or 2x = π or 2x = 2π
x = 0, x = π/2, x = π
the period of sin 2x is π, so adding multiples of π will yield more answers
so in the domain 0 < x < 2π , we have
x = 0, π/2, π, 3π/2, and 2π
(or x = 0, 90, 180, 270, 360 degrees)
BUT, since we squared our equation, all answers must be checked.
A quick check with a calculator shows that
x = 0, π/2, 2π are the only solutions
verfication:
https://www.wolframalpha.com/input/?i=cos(x)+%2B+sin(x)+%3D+1+for+x+%3D+0+to+2pi