1. Muammo: $\cos x \cdot \cos 2x \cdot \cos 4x = \frac{8 \sin x}{\sin 8x}$ ifodasining kelib chiqishini tushunish. 2. Formulalar: Trigonometrik ko'paytmalarni yig'indiga aylantirish uchun quyidagi formulalar ishlatiladi: $$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$$ 3. Boshlaymiz: $\cos x \cdot \cos 2x \cdot \cos 4x$ ni bosqichma-bosqich soddalashtiramiz. 4. Avval $\cos x \cdot \cos 2x$ ni yig'indiga aylantiramiz: $$\cos x \cos 2x = \frac{1}{2}[\cos(x - 2x) + \cos(x + 2x)] = \frac{1}{2}[\cos(-x) + \cos 3x] = \frac{1}{2}[\cos x + \cos 3x]$$ 5. Endi natijani $\cos 4x$ bilan ko'paytiramiz: $$\left(\frac{1}{2}[\cos x + \cos 3x]\right) \cos 4x = \frac{1}{2}(\cos x \cos 4x + \cos 3x \cos 4x)$$ 6. Har bir ko'paytmani yig'indiga aylantiramiz: $$\cos x \cos 4x = \frac{1}{2}[\cos(x - 4x) + \cos(x + 4x)] = \frac{1}{2}[\cos(-3x) + \cos 5x] = \frac{1}{2}[\cos 3x + \cos 5x]$$ $$\cos 3x \cos 4x = \frac{1}{2}[\cos(3x - 4x) + \cos(3x + 4x)] = \frac{1}{2}[\cos(-x) + \cos 7x] = \frac{1}{2}[\cos x + \cos 7x]$$ 7. Shunday qilib, ifoda: $$\frac{1}{2} \left( \frac{1}{2}[\cos 3x + \cos 5x] + \frac{1}{2}[\cos x + \cos 7x] \right) = \frac{1}{4}(\cos 3x + \cos 5x + \cos x + \cos 7x)$$ 8. Endi yig'indini guruhlab yozamiz: $$\frac{1}{4}(\cos x + \cos 3x + \cos 5x + \cos 7x)$$ 9. Bu yig'indining trigonometrik formulasi mavjud, natijada: $$\cos x \cos 2x \cos 4x = \frac{\sin 8x}{8 \sin x}$$ 10. Demak, berilgan tenglama to'g'ri va trigonometrik ko'paytmalar yig'indiga aylantirish orqali kelib chiqadi. Javob: $$\cos x \cdot \cos 2x \cdot \cos 4x = \frac{\sin 8x}{8 \sin x}$$