sin(2A+пи/4) = sin(2A) cos(π/4) + cos(2A) sin(π/4)

Зная, что sin(2A) = 2 sin(A) cos(A) и cos(2A) = cos^2(A) - sin^2(A), подставляем:

sin(2A+пи/4) = 2sin(A)cos(A)cos(π/4) + (cos^2(A) - sin^2(A))sin(π/4)
sin(2A+пи/4) = 2sin(A)cos(A)(√2/2) + (cos^2(A) - sin^2(A))(√2/2)
sin(2A+пи/4) = sin(2A) / √2 + (cos^2(A) - sin^2(A)) / √2
sin(2A+пи/4) = (2sin(A)cos(A) / √2) + ((cos^2(A) - sin^2(A)) / √2)

Пользуясь формулами для sin(2A) и cos(2A):

sin(2A+пи/4) = (√2sin(2A) + cos(2A)√2) / 2
sin(2A+пи/4) = √2*sin(2A + π/4) / 2

Итак, sin(2A+пи/4) = √2*sin(2A+π/4) / 2