To simplify the expression ctg(a) * sin(a) / (1 - (sin(a) + cos(a))^2, we can first expand the denominator and simplify the expression.

ctg(a) is equivalent to 1/tan(a), so the expression becomes:

1/tan(a) * sin(a) / (1 - (sin(a) + cos(a))^2

Next, expand the square of the sum in the denominator:

1/tan(a) * sin(a) / (1 - sin^2(a) - 2sin(a)cos(a) - cos^2(a))

Now, simplify the expression in the denominator by using the trigonometric identity sin^2(a) + cos^2(a) = 1:

1/tan(a) * sin(a) / (1 - 1 - 2sin(a)cos(a))

Further simplify by combining like terms:

1/tan(a) * sin(a) / (-2sin(a)cos(a))

Now, simplify by dividing sin(a) by cos(a) in the numerator:

= cot(a) * (sin(a) / cos(a)) / (-2)

= cot(a) * tan(a) / (-2)

= -cot(a) / 2

Therefore, ctg(a) * sin(a) / (1 - (sin(a) + cos(a))^2 simplifies to -cot(a) / 2.