To simplify the given expression:

ctg$2a$ - cos$2a$ / tg$2a$ - sin$2a$

First, we will use the double angle identities to simplify cosine and sine terms:

ctg$2a$ - cos$2a$ / tg$2a$ - sin$2a$ = ctg$2a$ - $co{s}^2(a)-si{n}^2(a)$ / tg$2a$ - 2sin$a$cos$a$ = ctg$2a$ - cos^2$a$ + sin^2$a$ / tg$2a$ - 2sin$a$cos$a$

Next, we will use the fact that the cotangent function is the reciprocal of the tangent function, and the Pythagorean trigonometric identity:

= 1/tg$2a$ - $1-si{n}^2(a)$ / tg$2a$ - 2sin$a$cos$a$ = 1/tg$2a$ - cos^2$a$ / tg$2a$ - 2sin$a$cos$a$

Now, we will replace the tangent and cotangent functions with sine and cosine functions:

= 1/$sin(2a)/cos(2a)$ - cos^2$a$ / $sin(2a)/cos(2a)$ - 2sin$a$cos$a$ = cos$2a$/sin$2a$ - cos^2$a$ / sin$2a$/cos$2a$ - 2sin$a$cos$a$ = cos^2$a$ - cos^2$a$ / sin^2$a$ - 2sin$a$cos$a$ = 0 / sin^2$a$ - 2sin$a$cos$a$ = 0 / sin$2a$ = 0

Therefore, the simplified expression is 0.