The value of `cot (underset(n=1)overset(23)sum cot^(-1) (1 + underset(k=1)overset(n)sum 2k))` is

Options

A. `(23)/(25)`
B. `(25)/(23)`
C. `(23)/(24)`
D. `(24)/(23)`

Answer

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

\[ \cot \left( \sum_{n=1}^{23} \cot^{-1} \left(1 + \sum_{k=1}^{n} 2k\right) \right) \]

### Step 1: Simplify the inner summation
The inner summation \( \sum_{k=1}^{n} 2k \) can be simplified using the formula for the sum of the first \( n \) natural numbers:

\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]

Thus,

\[ \sum_{k=1}^{n} 2k = 2 \cdot \frac{n(n+1)}{2} = n(n+1) \]

### Step 2: Substitute back into the cotangent expression
Now, substituting this back into our original expression gives:

\[ \cot \left( \sum_{n=1}^{23} \cot^{-1} (1 + n(n+1)) \right) \]

### Step 3: Rewrite the cotangent inverse
Using the identity \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \), we have:

\[ \cot^{-1}(1 + n(n+1)) = \tan^{-1}\left(\frac{1}{1 + n(n+1)}\right) \]

### Step 4: Sum the angles
Now we need to evaluate:

\[ \sum_{n=1}^{23} \tan^{-1}\left(\frac{1}{1 + n(n+1)}\right) \]

Using the identity for the difference of arctangents:

\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]

We can express:

\[ \tan^{-1}(n) - \tan^{-1}(n+1) = \tan^{-1}\left(\frac{n - (n+1)}{1 + n(n+1)}\right) = \tan^{-1}\left(\frac{-1}{1 + n(n+1)}\right) \]

### Step 5: Apply the telescoping series
Thus, we can rewrite the sum as:

\[ \sum_{n=1}^{23} \left( \tan^{-1}(n) - \tan^{-1}(n+1) \right) \]

This is a telescoping series, which simplifies to:

\[ \tan^{-1}(1) - \tan^{-1}(24) \]

### Step 6: Evaluate the cotangent of the result
Now we have:

\[ \cot\left(\tan^{-1}(1) - \tan^{-1}(24)\right) \]

Using the cotangent subtraction formula:

\[ \cot(A - B) = \frac{\cot A \cot B + 1}{\cot A - \cot B} \]

Where \( A = \tan^{-1}(1) \) (which gives \( \cot A = 1 \)) and \( B = \tan^{-1}(24) \) (which gives \( \cot B = 24 \)):

\[ \cot\left(\tan^{-1}(1) - \tan^{-1}(24)\right) = \frac{1 \cdot 24 + 1}{1 - 24} = \frac{25}{-23} = -\frac{25}{23} \]

### Final Result
Thus, the value of the original expression is:

\[ \frac{25}{23} \]