Ctg2 2x - 6 ctg 2x + 5 = 0

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To solve this equation, we can first simplify it by factoring out a common factor:

ctg 2x(2 - 6ctg 2x) + 5 = 0

Now, we can set each factor equal to zero:

ctg 2x = 0 and 2 - 6ctg 2x = 0

Solving the first equation, ctg 2x = 0, we know that cotangent is equal to zero at values of x where the tangent function is undefined (i.e. at multiples of pi). Therefore, the solutions for this equation are x = n*pi, where n is an integer.

Solving the second equation, 2 - 6ctg 2x = 0, we can rearrange to get:

6ctg 2x = 2
ctg 2x = 2/6
ctg 2x = 1/3

Now, we know that cotangent is equal to 1/3 at values of x where the tangent function is equal to 3. This occurs at approximately x = 0.98279 radians.

So, the solutions to the equation ctg 2x - 6 ctg 2x + 5 = 0 are x = npi and x ≈ 0.98279 + npi, where n is an integer.