To simplify the expression, let's first rewrite the terms using trigonometric identities.

tg(n/7) = sin(n/7) / cos(n/7)
ctg(n/7) = cos(n/7) / sin(n/7)

Now, we can rewrite the expression as:

sin(n/7) / cos(n/7) * cos(n/7) / sin(n/7) - sin(n/6)

Next, we simplify:

= sin(n/7) cos(n/7) / (cos(n/7) sin(n/7)) - sin(n/6)
= 1 - sin(n/6)

Therefore, the simplified expression is 1 - sin(n/6).