1) cos^2a+sin^2a+tg^2b 2) sin2a/2sina 3) sin a cos 3a - cos a sin 3a 4) (1-tgx) ^2 + (1+tgx) ^2

1) cos^2 (a) + sin^2 (a) + tg^2 (b) = 1/cos^2 (b)

a) По основному тригонометрическому тождеству имеем: sin^2 (a) + cos^2 (a) = 1, тогда cos^2 (a) + sin^2 (a) = 1

b) 1 + tg^2 (b) = 1/cos^2 (b)

2) sin (2a) / 2sin (a) = cos (a)

а) sin (2a) = 2 * sin (a) * cos (a)

b) sin (2a) / 2sin (a) = 2 * sin (a) * cos (a) / 2sin (a) = cos (a)

3) sin (a) * cos (3a) - cos (a) * sin (3a) = - 2 * sin (a) * cos (a)

a) по формуле sin (a - b) = sin (a) * cos (b) - cos (a) * sin (b)

b) sin (a) * cos (3a) - cos (a) * sin (3a) = sin (a - 3a) = sin (-2a) = - sin (2a) = - 2 * sin (a) * cos (a)

4) (1 - tg (x)) ^2 + (1 + tg (x)) ^2 = 1 - 2 * tg (x) + tg^2 (x) + 1 + 2 * tg (x) + tg^2 (x) = 1 + tg^2 (x) + 1 + tg^2 (x) = 2 (1 + tg^2 (x)) = 2 * 1/cos^2 (x) = 2/cos^2 (x)