Q.The value of $\cos 1^\circ\,\cos 2^\circ\,\cos 3^\circ \ldots \cos 179^\circ$ is
(A) $\dfrac{1}{\sqrt{2}}$
(B) $0$
(C) $1$
(D) $-1$

The key idea here is understanding the value of trigonometric functions at specific angles, particularly $\cos 90^\circ$.

The given expression is a product of cosine terms:

$\cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdot \ldots \cdot \cos 179^\circ$.

This product includes all integer angles from $1^\circ$ to $179^\circ$.

One of these angles is $90^\circ$.

We know that $\cos 90^\circ = 0$. …

When you multiply a series of numbers, if even one of those numbers is zero, the entire product collapses to zero. This fundamental property of multiplication is crucial here. We need to examine the terms in the given product to see if any of them evaluate to zero.

The expression is a product of cosine values for angles starting from $1^\circ$ and going up to $179^\circ$. We are looking for any angle in this sequence whose cosine is $0$.

1. Identify the terms in the product:

   The given expression is $P = \cos 1^\circ\,\cos 2^\circ\,\cos 3^\circ \ldots \cos 179^\circ$.

   This is a product of 179 terms, where each term is $\cos \theta$ for $\theta \in \{1^\circ, 2^\circ, \ldots, 179^\circ\}$.

2. Locate a special angle:

   We need to check if any angle in this range, when its cosine is taken, results in zero. A key angle for the cosine function is $90^\circ$.

3. Determine the value of $\cos 90^\circ$:

   The value of $\cos 90^\circ$ is $0$. This is a standard trigonometric value that should be memorized.

   $\cos 90^\circ = 0$

4. Confirm presence of $\cos 90^\circ$ in the product: …