How do you solve cos^2 x = 1 + sinx in the interval (0, 2π) ?

Use the identity that

sin² (x) + cos² (x) = 1

so

cos² (x) = 1+sin (x)

cos² (x) = sin² (x) + cos² (x) + sin (x)

minus cos² (x) both sides

0=sin² (x) + sin (x)

thefor

sin² (x) = - sin (x)

only possible if sin (x) = 0

or if sin (x) = - 1

that occurs at x=0 and x=3pi/2

but parethasees so 0 is not included

x=3pi/2