Ctg*x+tg*x=2.....* квадрат

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To solve this equation, we first need to substitute the values of cosecant and tangent in terms of sine and cosine.

Given equation: cot(x) + tan(x) = 2
Since cot(x) = 1/tan(x), let's substitute cot(x) in terms of tangent:
1/tan(x) + tan(x) = 2
Now, let's simplify the equation by getting a common denominator:
(1 + tan^2(x))/(tan(x)) = 2

Since 1 + tan^2(x) = sec^2(x), let's substitute sec^2(x) in terms of sine and cosine:
(sec^2(x))/(tan(x)) = 2

Now, let's simplify further by substituting sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x):
((1/cos^2(x))/(sin(x)/cos(x))) = 2
(1/cos(x))/(sin(x)/cos(x)) = 2
(1/cos(x))/(sin(x)) = 2
(1/(cos(x) sin(x))) = 2
1 = 2(cos(x) sin(x))

Since 1 = sin^2(x) + cos^2(x), let's substitute sin^2(x) in terms of cos(x):
1 = 2(cos(x) * sqrt(1 - cos^2(x)))

Now, let's square both sides to get rid of the square root:
1 = 4cos^2(x)(1 - cos^2(x))
1 = 4cos^2(x) - 4cos^4(x)

Rearranging the equation:
4cos^4(x) - 4cos^2(x) + 1 = 0

This is a quadratic equation in terms of cos(x). To solve for the values of cos(x), we can use the quadratic formula:
cos(x) = [4 ± sqrt((-4)^2 - 4(4)(1))] / (2*4)
cos(x) = [4 ± sqrt(16 - 16)] / 8
cos(x) = [4 ± 0] / 8
cos(x) = 1 or cos(x) = 1/2

Since cos(x) = 1 is not possible, the valid solution is:
cos(x) = 1/2

Now, we can find the value of sin(x):
sin(x) = sqrt(1 - cos^2(x))
sin(x) = sqrt(1 - (1/2)^2)
sin(x) = sqrt(1 - 1/4)
sin(x) = sqrt(3)/2

Therefore, the solutions for x are:
x = π/3 + 2πn, where n is an integer, and
x = 2π/3 + 2πn, where n is an integer.