If `(1-cot 1^(@))*(1-cot2^(@))*(1-cot3^(@))...(1-cot44^(@))=2^(n)`, then `(n)/(11)` is equal to

Explanation

To solve the problem, we need to evaluate the product:

\[ (1 - \cot 1^\circ)(1 - \cot 2^\circ)(1 - \cot 3^\circ) \cdots (1 - \cot 44^\circ) \]

We can use the property of cotangent:

\[ \cot(90^\circ - x) = \tan x \]

This means:

\[ \cot 44^\circ = \tan 46^\circ, \quad \cot 43^\circ = \tan 47^\circ, \quad \ldots, \quad \cot 1^\circ = \tan 89^\circ \]

Now, we can pair the terms in the product:

\[ (1 - \cot 1^\circ)(1 - \cot 44^\circ), \quad (1 - \cot 2^\circ)(1 - \cot 43^\circ), \quad \ldots, \quad (1 - \cot 22^\circ)(1 - \cot 23^\circ) \]

Each pair can be expressed as:

\[ (1 - \cot x)(1 - \cot(45^\circ - x)) = (1 - \cot x)(1 - \tan x) \]

Using the identity \( \cot x = \frac{1}{\tan x} \), we rewrite \( 1 - \cot x \):

\[ 1 - \cot x = 1 - \frac{1}{\tan x} = \frac{\tan x - 1}{\tan x} \]

Thus, the product becomes:

\[ \prod_{x=1}^{22} \left( (1 - \cot x)(1 - \cot(45^\circ - x)) \right) = \prod_{x=1}^{22} \left( \frac{\tan x - 1}{\tan x} \cdot \frac{\tan(45^\circ - x) - 1}{\tan(45^\circ - x)} \right) \]

Now, we can simplify this product further. Notice that:

\[ \tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x} \]

We can compute the product of \( (1 - \tan x)(1 - \tan(45^\circ - x)) \) for each \( x \) from 1 to 22.

After simplifying, we find that the entire product yields a power of 2. Specifically, it can be shown that:

\[ (1 - \cot 1^\circ)(1 - \cot 2^\circ) \cdots (1 - \cot 44^\circ) = 2^{22} \]

Thus, we have:

\[ 2^{n} = 2^{22} \implies n = 22 \]

Finally, we need to find \( \frac{n}{11} \):

\[ \frac{n}{11} = \frac{22}{11} = 2 \]

Therefore, the final answer is:

\[ \frac{n}{11} = 2 \]