To simplify the expression, we need to use trigonometric identities.

Given: U = 10ctg(x) + 5cos(x) + tg(x)

First, we know that cotangent is the reciprocal of tangent: ctg(x) = 1/tan(x).

So, U = 10(1/tan(x)) + 5cos(x) + tan(x)

Now, let's simplify:

U = 10/tan(x) + 5cos(x) + tan(x)

Next, we know that tan(x) = sin(x)/cos(x) and cos(x) = cos(x). Therefore, we can replace tan(x) and cos(x) with these expressions:

U = 10/(sin(x)/cos(x)) + 5cos(x) + sin(x)/cos(x)

U = 10cos(x)/sin(x) + 5cos(x) + sin(x)/cos(x)

Next, we can simplify further by finding a common denominator:

U = 10(cos^2(x)+5sin(x)cos(x)+sin^2(x))/sin(x)cos(x)

Therefore, the simplified expression is:

U = (10cos^2(x) + 5sin(x)cos(x) + sin^2(x)) / (sin(x)cos(x))