5. Докажите равенство 1/(2 sin 50°) + 2 sin 10° = 1.

1/(2 sin 50°) + 2 sin 10° = 1/(2 cos 40°) + 2 sin 10°

= (1 + 2 sin 10° cos 40°) / (2 cos 40°)

= (1 + sin 50° + sin(-30°)) / (2 cos 40°) = (1 + sin 50° - 1/2) / (2 cos 40°) = (1/2 + sin 50°) / (2 cos 40°)

= (1/2 + cos 40°) / (2 cos 40°) = 1/2 / (2 cos 40°) + 1/2 = 1/(4 cos 40°) + 1/2. This does not seem to lead to 1.

Let's try another approach: 1/(2 sin 50°) + 2 sin 10° = 1/(2 cos 40°) + 2 sin 10° = (1 + 4 sin 10° cos 40°)/(2 cos 40°) = (1 + 2(sin 50° - sin 30°))/(2 cos 40°) = (1 + 2 sin 50° - 1)/(2 cos 40°) = (2 sin 50°)/(2 cos 40°) = sin 50°/cos 40° = cos 40°/cos 40° = 1.

Доказано.