## Answer :

1.

**Rewrite the equation with trigonometric identities**:

Recall that [tex]\( \cos^2(x) = 1 - \sin^2(x) \)[/tex]. Using this identity, we rewrite the equation:

[tex]\[ 1 - \sin^2(x) = 2 + 2\sin(x) \][/tex]

2.

**Form a quadratic equation**:

Rearrange the equation to bring all terms to one side:

[tex]\[ 1 - \sin^2(x) - 2\sin(x) - 2 = 0 \][/tex]

Simplify it:

[tex]\[ -\sin^2(x) - 2\sin(x) - 1 = 0 \][/tex]

Multiply by [tex]\(-1\)[/tex] to make it easier to solve:

[tex]\[ \sin^2(x) + 2\sin(x) + 1 = 0 \][/tex]

3.

**Solve the quadratic equation**:

Let [tex]\( u = \sin(x) \)[/tex]. The quadratic equation becomes:

[tex]\[ u^2 + 2u + 1 = 0 \][/tex]

Factorize the quadratic equation:

[tex]\[ (u + 1)^2 = 0 \][/tex]

So, we have:

[tex]\[ u + 1 = 0 \][/tex]

Solving for [tex]\( u \)[/tex]:

[tex]\[ u = -1 \][/tex]

4.

**Find the corresponding angle for [tex]\( \sin(x) = -1 \)[/tex]**:

We need to find [tex]\( x \)[/tex] such that [tex]\( \sin(x) = -1 \)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]. The value [tex]\( \sin(x) = -1 \)[/tex] occurs at:

[tex]\[ x = \frac{3\pi}{2} \][/tex]

Therefore, the solution to the equation [tex]\( \cos^2(x) = 2 + 2\sin(x) \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] is:

[tex]\[ x = \frac{3\pi}{2} \][/tex]