To solve this equation, we can first simplify the left side by factoring out a common factor of tg3x:

tg4x - tg3x + tg4x * tg3x = √3
tg3x(tg4x - 1) + tg4x = √3

Now we can substitute tg3x = a and tg4x = b to make the equation easier to solve:

a(b - 1) + b = √3

Expanding the left side, we get:

ab - a + b = √3

Now we can substitute back in tg3x and tg4x:

tg3x * tg4x - tg3x + tg4x = √3

Finally, we have the simplified equation:

tg4x * tg3x - tg3x + tg4x = √3

This is the final solution to the given equation.