To solve the given equations tg(X) - 3ctg(X) = 0 and sin(3X7) - sin(X) = 0, we need to simplify and rewrite the equations in a more manageable form.

For the first equation:
tg(X) - 3ctg(X) = 0
tg(X) - 3(1/tg(X)) = 0
tg(X) - 3/tg(X) = 0
(tg^2(X) - 3) / tg(X) = 0

Since the numerator must be equal to zero for the fraction to be zero, we have:
tg^2(X) - 3 = 0
tg^2(X) = 3
tg(X) = ±√3
X = arctg(√3) or X = arctg(-√3)

For the second equation:
sin(3X) - sin(X) = 0

We can use the trigonometric identity for the sine of the difference of angles:
sin(3X) - sin(X) = 2sin((3X - X)/2)cos((3X + X)/2) = 0
sin(2X)cos(2X) = 0

Now, we have two cases to consider for the solution:

sin(2X) = 0, which implies that 2X = kπ where k is an integer.cos(2X) = 0, which implies that 2X = (2n+1)π/2 where n is an integer.

Combining the solutions from both cases, we have:
X = kπ/2 or X = (2n+1)π/4.

These are the solutions to the given equations tg(X) - 3ctg(X) = 0 and sin(3X) - sin(X) = 0.