Tg^2(x)+ctg^2(x)+3tg(x)+3ctg(x)=-4

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To solve this equation, let's first rewrite it in terms of sine and cosine functions:

Given:
tg(x) = sin(x)/cos(x) = sqrt(1-cos^2(x))/cos(x) = sqrt((1-cos(x))(1+cos(x)))/cos(x)
ctg(x) = cos(x)/sin(x) = cos(x)/sqrt(1-cos^2(x))

Now the equation becomes:
(sqrt((1-cos(x))(1+cos(x)))/cos(x))^2 + (cos(x)/sqrt(1-cos^2(x)))^2 + 3(sqrt((1-cos(x))(1+cos(x)))/cos(x)) + 3(cos(x)/sqrt(1-cos^2(x))) = -4

Expand and simplify:
(1-cos(x))(1+cos(x)) + cos^2(x) + (3(1-cos(x))(1+cos(x))) + 3cos^2(x) = -4

Simplify further:
1 - cos^2(x) + cos^2(x) + 3 - 3cos^2(x) + 3cos^2(x) = -4

1 + 3 = -4
4 = -4

This is not a valid equation, as 4 is not equal to -4. Therefore, the original equation has no solution.