What is the value of $\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \cos 3^{\circ }\dots \cos 177^{\circ }\cdot \cos 178^{\circ }\cdot \cos 179^{\circ }$

The given expression is a continuous product of cosine values from $1^{\circ }$ to $179^{\circ }$:
$$P=\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \cos 3^{\circ }\dots \cos 90^{\circ }\dots \cos 178^{\circ }\cdot \cos 179^{\circ }$$
Within this sequence, the term $\cos 90^{\circ }$ is included.We know that the exact value of $\cos 90^{\circ }=0$.Since any number multiplied by zero is zero, the entire product becomes zero:
$$P=\cos 1^{\circ }\cdot \cos 2^{\circ }\cdots \times 0\times \dots \cos 179^{\circ }=0$$