To solve this equation, we can first simplify the expression on the left side:

tgx/1 + tg2x(1 + ctg2x) = 1
=> tgx + tg2x + tg2x ctg2x = 1

Next, we can use the trigonometric identity:
tg2x * ctg2x = 1

Therefore, our equation simplifies to:
tgx + tg2x + 1 = 1

From here, we can see that tg2x + 1 must be equal to 0 for the equation to hold true. This means that tg2x = -1.

Now, to solve for x, we can equate tg2x to -1:
tg(2x) = -1
=> 2x = arctan(-1)
=> 2x = -π/4
=> x = -π/8

Therefore, the solution to the equation is x = -π/8.