Find the value of cos `1^@ cos 2^@ cos 3^@` …….cos `89^@`…..cos `179^@ cos 180^@`

Answer

To find the value of the product \( \cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdots \cos 89^\circ \cdots \cos 179^\circ \cdot \cos 180^\circ \), we can break down the problem step by step.

### Step 1: Identify the range of angles
The product consists of cosines from \( \cos 1^\circ \) to \( \cos 180^\circ \). Notably, \( \cos 180^\circ = -1 \).

### Step 2: Pair the cosines
We can pair the cosines symmetrically around \( 90^\circ \):
- \( \cos 1^\circ \) pairs with \( \cos 179^\circ \)
- \( \cos 2^\circ \) pairs with \( \cos 178^\circ \)
- ...
- \( \cos 89^\circ \) pairs with \( \cos 91^\circ \)

### Step 3: Use the cosine identity
Using the identity \( \cos(180^\circ - x) = -\cos x \), we can see that:
- \( \cos 1^\circ \cdot \cos 179^\circ = \cos 1^\circ \cdot (-\cos 1^\circ) = -\cos^2 1^\circ \)
- \( \cos 2^\circ \cdot \cos 178^\circ = \cos 2^\circ \cdot (-\cos 2^\circ) = -\cos^2 2^\circ \)
- ...
- \( \cos 89^\circ \cdot \cos 91^\circ = \cos 89^\circ \cdot (-\cos 89^\circ) = -\cos^2 89^\circ \)

### Step 4: Include \( \cos 90^\circ \)
Since \( \cos 90^\circ = 0 \), the entire product will be zero because multiplying by zero results in zero.

### Conclusion
Thus, the value of the product \( \cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdots \cos 89^\circ \cdots \cos 179^\circ \cdot \cos 180^\circ \) is:

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