Question

Cos 12° cos 24° cos 48° cos 96°​

Answer 1

Смотри.......................

Answer 2

To calculate the value of the expression $\cos(12^\circ) \cos(24^\circ) \cos(48^\circ) \cos(96^\circ)$, you can use the trigonometric identity:

$\cos(A) \cos(B) = \frac{1}{2} \left[ \cos(A + B) + \cos(A - B) \right]$

So, let's proceed with the calculation step by step:

Step 1: Calculate $\cos(12^\circ + 24^\circ)$ and $\cos(12^\circ - 24^\circ)$ $\cos(12^\circ + 24^\circ) = \cos(36^\circ)$ $\cos(12^\circ - 24^\circ) = \cos(-12^\circ) = \cos(12^\circ)$

Step 2: Calculate $\cos(36^\circ + 48^\circ)$ and $\cos(36^\circ - 48^\circ)$ $\cos(36^\circ + 48^\circ) = \cos(84^\circ)$ $\cos(36^\circ - 48^\circ) = \cos(-12^\circ) = \cos(12^\circ)$

Step 3: Calculate $\cos(84^\circ + 96^\circ)$ and $\cos(84^\circ - 96^\circ)$ $\cos(84^\circ + 96^\circ) = \cos(180^\circ) = -1$ $\cos(84^\circ - 96^\circ) = \cos(-12^\circ) = \cos(12^\circ)$

Now, let's combine the results: $\cos(12^\circ) \cos(24^\circ) \cos(48^\circ) \cos(96^\circ) = \frac{1}{2} \left[ \cos(36^\circ) + \cos(12^\circ) \right] \times \frac{1}{2} \left[ \cos(84^\circ) + \cos(12^\circ) \right] \times (-1)$

Substitute the calculated values: $\cos(12^\circ) \cos(24^\circ) \cos(48^\circ) \cos(96^\circ) = \frac{1}{2} \left[ \cos(36^\circ) + \cos(12^\circ) \right] \times \frac{1}{2} \left[ \cos(84^\circ) + \cos(12^\circ) \right] \times (-1)$ $= \frac{1}{2} \left[ \cos(36^\circ) + \cos(12^\circ) \right] \times \frac{1}{2} \left[ \cos(84^\circ) + \cos(12^\circ) \right] \times (-1)$ $= \frac{1}{2} \left[ \cos(36^\circ) + \cos(12^\circ) \right] \times \frac{1}{2} \left[ -1 + \cos(12^\circ) \right]$