To simplify the expression, we can use trigonometric identities:

So, ctg²β = 1 / cot²β = sin²β / cos²β

Now, let's substitute these identities into the expression:

tg²β - sin²β : ctg²β - cos²β
= (sin²β / cos²β) - sin²β : (sin²β / cos²β) - cos²β
= (sin²β - sin²β cos²β) / cos²β : (sin²β - cos²β sin²β) / sin²β
= sin²β / cos²β : - cos²β / sin²β
= tan²β : - cot²β

Thus, the simplified expression is tan²β : - cot²β.