1) To simplify the expression cos a/1-2sin^2 a + sin a/1-2cos^2 a, we can first factor out a common factor of 1/1-2sin^2 a from the first term and a common factor of 1/1-2cos^2 a from the second term:

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

Next, we can use the identity sin^2 a + cos^2 a = 1 to rewrite the denominators as:

cos a/(cos^2 a) + sin a/(sin^2 a)

Now, we can simplify the expression to obtain:

tan a + ctg a

2) To simplify the expression 1/cos^2 a - 1/ctg^2 a - sin^2 a/tg^2 a, we can rewrite ctg a as 1/tg a:

1/cos^2 a - tg^2 a - sin^2 a/tg^2 a

Next, we can rewrite tg a as sin a/cos a and simplify the expression further:

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

1/cos^2 a - (1) - sin^4 a/(cos^2 a)(sin^2 a)

1/cos^2 a - 1 - sin^2 a/cos^2 a

3) To simplify the expression (ctg a - cos a)(sin^2 a/cos a + tg a), we can first expand the terms:

ctg a(sin^2 a/cos a + tg a) - cos a(sin^2 a/cos a + tg a)

Next, we can distribute the terms:

ctg a(sin a - cos a) + ctg^2 a - cos asin^2 a - cos atg a

ctg asin a - ctg acos a + 1 - sin^2 a - sin a

Finally, we can simplify the expression to obtain:

1 - sin a*cos a