Recognize Pythagorean Identity: Given the equation cos^(2)x + 2sin x = -2, we need to find all values of x that satisfy this equation. First, we recognize that cos^(2)x can be written as 1 - sin^(2)x using the Pythagorean identity. So, we rewrite the equation as: 1 - sin^(2)x + 2sin x = -2Simplify Equation: Now, we simplify the equation by moving all terms to one side to set the equation to zero: -sin^(2)x + 2sin x + 3 = 0 This is a quadratic equation in terms of sin x.Factor Quadratic Equation: To solve the quadratic equation, we can factor it if possible. Let's try to factor the equation: (-sin x + 3)(sin x + 1) = 0Set Factors to Zero: We now have two factors that can be set to zero to find the solutions for sin x: -sin x + 3 = 0 or sin x + 1 = 0 Solving these, we get sin x = 3 or sin x = -1.Check Valid Solutions: We check the solutions for any mathematical errors. Since the sine function has a range of [-1, 1], sin x = 3 is not possible. Therefore, the only valid solution from the factors is sin x = -1.Find Sine Values: We find the values of x that satisfy sin x = -1. The sine function is equal to -1 at x = 3π/2 + 2πn, where n is an integer.Conclude Valid Values: We conclude that the only values of x that satisfy the original equation are x = 3π/2 + 2πn, where n is an integer.

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$\cos ^{2} x+2 \sin x=-2$

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\cos ^{2} x+2 \sin x=-2

Recognize Pythagorean Identity: Given the equation $\cos^2 x + 2\sin x = -2$ , we need to find all values of $x$ that satisfy this equation. $\newline$ First, we recognize that $\cos^2 x$ can be written as $1 - \sin^2 x$ using the Pythagorean identity. $\newline$ So, we rewrite the equation as: $\newline$ $1 - \sin^2 x + 2\sin x = -2$
Simplify Equation: Now, we simplify the equation by moving all terms to one side to set the equation to zero: $\newline$ $-\sin^{2}x + 2\sin x + 3 = 0$ $\newline$ This is a quadratic equation in terms of $\sin x$ .
Factor Quadratic Equation: To solve the quadratic equation, we can factor it if possible. Let's try to factor the equation: $\newline$ $(-\sin x + 3)(\sin x + 1) = 0$
Set Factors to Zero: We now have two factors that can be set to zero to find the solutions for $\sin x$ : $-\sin x + 3 = 0$ or $\sin x + 1 = 0$ Solving these, we get $\sin x = 3$ or $\sin x = -1$ .
Check Valid Solutions: We check the solutions for any mathematical errors. Since the sine function has a range of $[-1, 1]$ , $\sin x = 3$ is not possible. $\newline$ Therefore, the only valid solution from the factors is $\sin x = -1$ .
Find Sine Values: We find the values of $x$ that satisfy $\sin x = -1$ . The sine function is equal to $-1$ at $x = 3\pi/2 + 2\pi n$ , where $n$ is an integer.
Conclude Valid Values: We conclude that the only values of $x$ that satisfy the original equation are $x = \frac{3\pi}{2} + 2\pi n$ , where $n$ is an integer.