Question

Докажите тождество a) (sina+cosa)2=1+sin2a б) 2sina=1-cos2a

Answer 1

a) (sina+cosa)^2=sin^2a +2*sina*cosa+cos^2a= (sin^2a+cos^2a +2*sina*cosa=1+sin2a.  Такое же выражение в правой части.

Answer 2

Доказательство тождества (а):

Для доказательства тождества (a): (sin(a) + cos(a))^2 = 1 + sin^2(a), мы начнем с левой стороны и постепенно преобразуем ее, используя тригонометрические тождества.

Левая сторона:

(a) (sin(a) + cos(a))^2

(a) (sin^2(a) + 2sin(a)cos(a) + cos^2(a))

(a) (1 - cos^2(a) + 2sin(a)cos(a) + cos^2(a)) (используем тригонометрическое тождество sin^2(a) + cos^2(a) = 1)

(a) (1 + 2sin(a)cos(a))

(a) + 2sin(a)cos(a) (так как 1 + 2sin(a)cos(a) = 1 + sin^2(a))

Правая сторона:

1 + sin^2(a)

1 + sin^2(a) + 0sin(a)cos(a) (добавляем 0sin(a)cos(a), чтобы обеспечить равенство)

1 + sin^2(a) + sin(a)cos(a) - sin(a)cos(a) (добавляем и вычитаем sin(a)cos(a))

(1 + sin(a)cos(a)) + sin^2(a) - sin(a)cos(a)

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1 + sin(a)cos(a)) + sin(a)(sin(a) - cos(a))

(1