sin 10° · sin 50° · sin 70°.

Преобразуем произведение sin 10° \(\cdot\) sin 50°, используя формулу: sin α \(\cdot\) sin β = \(\frac{1}{2}\) [cos(α - β) - cos(α + β)] sin 10° \(\cdot\) sin 50° = \(\frac{1}{2}\) [cos(10° - 50°) - cos(10° + 50°)] = \(\frac{1}{2}\) [cos(-40°) - cos 60°] = \(\frac{1}{2}\) [cos 40° - \(\frac{1}{2}\)] Исходное выражение: \(\frac{1}{2}\) [cos 40° - \(\frac{1}{2}\)] \(\cdot\) sin 70° = \(\frac{1}{2}\) \(\cdot\) cos 40° \(\cdot\) sin 70° - \(\frac{1}{4}\) \(\cdot\) sin 70° Преобразуем произведение cos 40° \(\cdot\) sin 70°, используя формулу: cos α \(\cdot\) sin β = \(\frac{1}{2}\) [sin(α + β) + sin(β - α)] cos 40° \(\cdot\) sin 70° = \(\frac{1}{2}\) [sin(40° + 70°) + sin(70° - 40°)] = \(\frac{1}{2}\) [sin 110° + sin 30°] = \(\frac{1}{2}\) [sin 110° + \(\frac{1}{2}\)] Подставим обратно: \(\frac{1}{2}\) \(\cdot\) \(\frac{1}{2}\) [sin 110° + \(\frac{1}{2}\)] - \(\frac{1}{4}\) \(\cdot\) sin 70° = \(\frac{1}{4}\) sin 110° + \(\frac{1}{8}\) - \(\frac{1}{4}\) sin 70° Т.к. sin 110° = sin (180° - 70°) = sin 70°: \(\frac{1}{4}\) sin 70° + \(\frac{1}{8}\) - \(\frac{1}{4}\) sin 70° = \(\frac{1}{8}\) Ответ: \(\frac{1}{8}\)