Recall the double angle identity for cosine:
cos(x) = cos(2×x/2) = 1 - 2 sin²(x/2)
Then the equation can be rewritten as
sin(x/2) + (1 - 2 sin²(x/2)) - 1 = 0
sin(x/2) - 2 sin²(x/2) = 0
sin(x/2) (1 - 2 sin(x/2)) = 0
sin(x/2) = 0 or 1 - 2 sin(x/2) = 0
sin(x/2) = 0 or sin(x/2) = 1/2
[x/2 = arcsin(0) + 360n ° or x/2 = 180° - arcsin(0) + 360n °]
… … or [x/2 = arcsin(1/2) + 360n ° or x/2 = 180° - arcsin(1/2) + 360n °]
x/2 = 360n ° or x/2 = 180° + 360n °
… … or x/2 = 30° + 360n ° or x/2 = 150° + 360n °
x = 720n ° or x = 360° + 720n °
… … or x = 60° + 720n ° or x = 300° + 720n °
(where n is any integer)
We get only three solutions in 0° ≤ x < 360° :
720×0° = 0°
60° + 720×0° = 60°
300° + 720×0° = 300°