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) 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