cos2x-√3sinx=1 could mean:
(1) cos(2x)-(√3)sin(x)=1
(2) cos²(x)-(√3)sin(x)=1
(3) cos(2x)-√(3sin(x))=1
(4) cos²(x)-√(3sin(x))=1
I’ll assume (1), which can be written:
1-2sin²(x)=1+(√3)sin(x),
-2sin²(x)=(√3)sin(x),
sin(x)(√3+2sin(x))=0,
sin(x)=0, or sin(x)=-√3/2. Note that sin⁻¹(√3/2)=π/3 (60°).
In the interval [0,2π), x=0, x=sin⁻¹(-√3/2)=5π/3 (300°) or 4π/3 (240°).
This is the most likely interpretation.
(2) can be written:
1-sin²(x)=1+(√3)sin(x),
sin(x)(√3+sin(x))=0, x=0 is the only solution.
(3) and (4) lead to quartic equations in sin(x) so are unlikely interpretations.