Solving for x on unit circle equation

Question

I have been given the equation $$\cos^2{x} + 2\sin{x}=2.$$

I have factored it, and the only answer I got was $x=\frac{\pi}{2}$.

Is this correct or is there more than one answer? The interval is $0 \leq x \leq 2\pi$.

So it is withing one rotation of the unit circle.

Answer 1

$$1-\sin^2 x+2\sin x=2\implies \sin^2 x-2\sin x+1=0\implies (\sin x-1)^2=0$$ so $$\sin x=1$$ and the solutions are $${\pi\over 2}+2n\pi,\;\;n\in \mathbb{Z}$$

Answer 2

$$\cos^2x=1-\sin^2x\implies 1-\sin^2x+2\sin x=2\iff\sin^2x-2\sin x+1=0\iff$$

$$(\sin x-1)^2=0\iff \sin x=1\iff x=\frac\pi2+2k\pi\;,\;\;k\in\Bbb Z$$