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

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.