§ 25: 25.41

25.41 Доведіть, що:

1) 4sinαsin(60° – α)sin(60 + α) = 4sinα(sin60°cos α – cos60°sinα) • (sin60°cosα + cos60°sinα) = 4sinα(32cosα – 12sinα) = (32cosα + 12sinα) = 4sinα(34cos²α – 14sin²α) = 3sinαcos²α – sin²α = 3sinα(1 – sin²α) – sin³α = 3sinα – 3sinα – sin³α = 3sinα – 4sin³α = sin3α; sin3α = sin3α, тотожність доведено.

3) 4sin20°sin 50°sin70°sin 80°⁡⁡= 4sin20°sin70°-20°sin70°sin80° = 4sin20°sin70°cos20° - cos70°sin20°sin70°sin 80° = 4sin20°cos20°sin270° - 4sin220°cos70°sin70°sin80° = 2sin40°sin270° - 2sin20°sin140°sin80° = 2sin40°(cos220° - sin220°)sin80° = 2sin40°cos40°sin80° = 1.