We are tasked with finding the value of \( \cos 1^\circ \cos 2^\circ \cdots \cos 180^\circ \).
Step 1: Analyze the product.
The product involves all cosine values from \( \cos 1^\circ \) to \( \cos 180^\circ \). A key observation is that \( \cos 90^\circ = 0 \).
Step 2: Impact of \( \cos 90^\circ = 0 \).
Since one of the terms in the product is \( \cos 90^\circ = 0 \), the entire product becomes zero. This is because multiplying any number by zero results in zero.
Final Answer: The value of \( \cos 1^\circ \cos 2^\circ \cdots \cos 180^\circ \) is \( \mathbf{0} \).
Statement-I: In the interval \( [0, 2\pi] \), the number of common solutions of the equations
\[ 2\sin^2\theta - \cos 2\theta = 0 \]
and
\[ 2\cos^2\theta - 3\sin\theta = 0 \]
is two.
Statement-II: The number of solutions of
\[ 2\cos^2\theta - 3\sin\theta = 0 \]
in \( [0, \pi] \) is two.