Question

2 cos x - ctg x - 2 sin x + 1 = 0

Answer 1

2 * (cos x - sin x) - (cos x / sin x - 1) = 0

2 * sin x * (cos x - sin x) - (cos x - sin x)

------------------------------------------------ = 0

                       sin x

(2 * sin x - 1) * (cos x - sin x)

------------------------------------ = 0

                     sin x

1) 2 * sin x = 1                                   2)  cos x = sin x

         sin x = 0,5                                       tg x = 1

         x = (-1)^n * π/6 + π * n                    x = π / 4 + π * n

При этих значениях х знаменатель нулю не равен. поэтому все они подходят

Answer 2

To solve the equation 2 cos(x) - ctg(x) - 2 sin(x) + 1 = 0, we can use trigonometric identities and algebraic manipulations. Let's break it down step by step:

1. Convert cotangent (ctg) to its equivalent form using cosine and sine: ctg(x) = 1/tan(x) = cos(x)/sin(x)

2. Substitute ctg(x) with its equivalent form in the equation: 2 cos(x) - (cos(x)/sin(x)) - 2 sin(x) + 1 = 0

3. Multiply the entire equation by sin(x) to eliminate the denominator: 2 cos(x)sin(x) - cos(x) - 2 sin^2(x) + sin(x) = 0

4. Rearrange the terms and group them: 2 cos(x)sin(x) - 2 sin^2(x) - cos(x) + sin(x) = 0

5. Factor out common terms: sin(x)(2 cos(x) - 2 sin(x)) - (cos(x) - sin(x)) = 0

6. Factor out common factors in both terms: sin(x)(2(cos(x) - sin(x))) - (cos(x) - sin(x)) = 0

7. Now we have a common factor of (cos(x) - sin(x)): (cos(x) - sin(x))(sin(x) - 1) = 0

8. Set each factor equal to zero and solve for x: cos(x) - sin(x) = 0 => cos(x) = sin(x) sin(x) - 1 = 0 => sin(x) = 1

For the equation cos(x) = sin(x), we know that cos(x) = sin(x) only when x = π/4 + 2πn, where n is an integer.

For the equation sin(x) = 1, we know that sin(x) = 1 when x = π/2 + 2πn, where n is an integer.

So, the solutions to the given equation are: x = π/4 + 2πn, where n is an integer x = π/2 + 2πn, where n is an integer.