Доказать тождества: 1) (1-cos2a) (1+cos2a) = sin^2 2a 2) sin a-1/cos^2=-1/1+sina 3) cos^4a-sin^4a=cos^2a-sin^2 4) sina/1+cosa+1+cosa/sina=2/sina 5) sina/1-cosa=1+cosa/sina 6) 1/1+tg^a+1/1+ctg^a=1 7) tg^2a-sin^2a=th^2a sin^2a

Докажем тождества:

1) (1 - cos (2 * a)) (1 + cos (2 * a)) = sin^2 (2 * a);

1^2 - cos^2 (2 * a) = sin^2 (2 * a);

1 - cos^2 (2 * a) = sin^2 (2 * a);

cos^2 (2 * a) + sin^2 (2 * a) - cos^2 (2 * a) = sin^2 (2 * a);

sin^2 (2 * a) = sin^2 (2 * a);

Значит, тождество верно.

2) (sin a - 1) / cos^2 а = - 1 / (1 + sin a);

(sin a - 1) / (1 - sin^2 a) = - 1 / (1 + sin a);

-1 / (1 + sin a) = - 1 / (1 + sin a);

Верно.

3) cos^4 a - sin^4 a = cos^2 a - sin^2 a;

(cos^2 a - sin^2 a) * (cos^2 a + sin^2 a) = cos^2 a - sin^2 a;

cos^2 a - sin^2 a = cos^2 a - sin^2 a;

Верно.

4) sin a / (1 + cos a) + (1 + cos a) / sin a = 2/sin a;

(sin^2 a + 1 + cos^2 a) / (sin a * (1 + cos a)) = 2/sin a;

(1 + 1) / (sin a * (1 + cos a)) = 2/sin a;

2 / (sin a * (1 + cos a)) = 2/sin a;

Тождество неверно.

5) sin a / (1 - cos a) = (1 + cos a) / sin a;

sin^2 a = 1 - cos^2 a;

sin^2 a = sin^2 a + cos^2 a - cos^2 a;

sin^2 a = sin^2 a;

Верно.

6) 1 / (1 + tg^2 a) + 1 / (1 + ctg^2 a) = 1;

(1 + ctg^2 a + 1 + tg^2 a) / ((1 + tg^2 a) * (1 + ctg^2 a) = 1;

(2 + ctg^2 a + tg^2 a) / ((1 + tg^2 a) * (1 + ctg^2 a) = 1;

(2 + ctg^2 a + tg^2 a) = ((1 + tg^2 a) * (1 + ctg^2 a);

(2 + ctg^2 a + tg^2 a) = 1 + ctg^2 a + tg^2 a + tg^2 a * ctg^2 a;

2 + ctg^2 a + tg^2 a = 1 + ctg^2 a + tg^2 a + 1;

2 + ctg^2 a + tg^2 a = 2 + ctg^2 a + tg^2 a;

Верно.

7) tg^2a - sin^2 a = tg^2a - sin^2 a;

sin^2a/cos^2 a - sin^2 a = tg^2a - sin^2 a;

(sin^2 a - sin^2 a * cos^2 a) / cos^2 a = tg^2a - sin^2 a;

(sin^2 a/cos^2 a - sin^2 a * cos^2 a/cos^2 a = tg^2a - sin^2 a;

tg^2 a - sin^2 a = tg^2 a - sin^2 a;

Верно.