Упростить уравнение: а) (tga+ctga) * (1+cosa) * (1-cosa) б) cos^3a+sin^3a/sina+cos^3a-sin^3a/cosa

а) (tg a + ctg a) * (1 + cos a) * (1 - cos a) = (tg a + ctg a) * (1^2 - cos^2 a) =

= (tg a + ctg a) * sin^2 a = [ (sin a) / (cos a) + (cos a) / (sin a) ] * sin^2 a =

= [ (sin^2 a) / (cos a * sin a) + (cos^2 a) / (sin a * cos a) ] * sin^2 a =

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

= (sin a) / (cos a) = tg a.

б) [cos^3 a + (sin^3 a) ] / (sin a) + [cos^3 a - (sin^3 a) ] / (cos a) =

= (cos^3 a) / (sin a) + sin^2 a + cos^2 a - (sin^3 a) / (cos a) =

= (cos^3 a) / (sin a) + 1 - (sin^3 a) / (cos a) =

= 1 + (cos^4 a) / (sin a * cos a) - (sin^4 a) / (sin a * cos a) =

= 1 + (cos^4 a - sin^4 a) / (sin a * cos a) =

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

= 1 + (cos^2 a - sin^2 a) / (sin a * cos a) = 1 + cos 2a / (0,5 * sin 2a) =

= 1 + 2 * cos 2a / sin 2a = 1 + 2 * ctg 2a.