Question

2ctg^2 x+3=3sin^-2 x

Answer 1

Решение во вложении.

Answer 2

To solve the equation 2ctg^2(x) + 3 = 3sin^(-2)(x), we'll need to manipulate the terms to isolate the trigonometric functions. First, let's recall some trigonometric identities:

1. cot^2(x) + 1 = csc^2(x)
2. csc^2(x) = 1/sin^2(x)

Now, let's proceed with the solution:

Step 1: Substitute cot^2(x) with (csc^2(x) - 1) in the equation. 2(csc^2(x) - 1) + 3 = 3/sin^2(x)

Step 2: Expand and simplify. 2csc^2(x) - 2 + 3 = 3/sin^2(x)

Step 3: Combine like terms. 2csc^2(x) + 1 = 3/sin^2(x)

Step 4: Now, rearrange the equation to isolate csc^2(x). 2csc^2(x) = 3/sin^2(x) - 1

Step 5: Find a common denominator on the right side. 2csc^2(x) = (3 - sin^2(x)) / sin^2(x)

Step 6: Apply the Pythagorean identity (sin^2(x) + cos^2(x) = 1). 2csc^2(x) = (3 - (1 - cos^2(x))) / sin^2(x) 2csc^2(x) = (2 + cos^2(x)) / sin^2(x)

Step 7: Now, rewrite csc^2(x) as 1/sin^2(x). 2/sin^2(x) = (2 + cos^2(x)) / sin^2(x)

Step 8: Cancel out sin^2(x) from both sides (since it is nonzero). 2 = 2 + cos^2(x)

Step 9: Now, isolate cos^2(x) on one side. cos^2(x) = 0

Step 10: Take the square root of both sides. cos(x) = ±√0

Step 11: Simplify and solve for x. cos(x) = 0

The solutions for x are the angles at which the cosine function is equal to zero. The two principal solutions within the interval [0, 2π] are π/2 and 3π/2.

So, the solutions for the given equation are: x = π/2 + kπ and x = 3π/2 + kπ, where k is an integer (positive, negative, or zero) representing additional solutions due to the periodic nature of trigonometric functions.