If `cosec . (pi)/(32)+ cosec. (pi)/(16)+ cosec. (pi)/(8)+ cosec. (pi)/(4)+ cosec. (pi)/(2)= cot. (pi)/(k)`, then the value of k is

Options

A. 64
B. 96
C. 48
D. 32

Answer

To solve the equation

\[ \csc\left(\frac{\pi}{32}\right) + \csc\left(\frac{\pi}{16}\right) + \csc\left(\frac{\pi}{8}\right) + \csc\left(\frac{\pi}{4}\right) + \csc\left(\frac{\pi}{2}\right) = \cot\left(\frac{\pi}{k}\right), \]

we will use the identity that relates cosecant and cotangent.

### Step 1: Rewrite each cosecant in terms of cotangent
We know that

\[ \csc \theta = \frac{1}{\sin \theta} = \cot\left(\frac{\theta}{2}\right) - \cot \theta. \]

Using this property, we can rewrite each term:

1. \(\csc\left(\frac{\pi}{32}\right) = \cot\left(\frac{\pi}{64}\right) - \cot\left(\frac{\pi}{32}\right)\)
2. \(\csc\left(\frac{\pi}{16}\right) = \cot\left(\frac{\pi}{32}\right) - \cot\left(\frac{\pi}{16}\right)\)
3. \(\csc\left(\frac{\pi}{8}\right) = \cot\left(\frac{\pi}{16}\right) - \cot\left(\frac{\pi}{8}\right)\)
4. \(\csc\left(\frac{\pi}{4}\right) = \cot\left(\frac{\pi}{8}\right) - \cot\left(\frac{\pi}{4}\right)\)
5. \(\csc\left(\frac{\pi}{2}\right) = \cot\left(\frac{\pi}{4}\right) - \cot\left(\frac{\pi}{2}\right)\)

Since \(\cot\left(\frac{\pi}{2}\right) = 0\), we can simplify:

\[ \csc\left(\frac{\pi}{2}\right) = \cot\left(\frac{\pi}{4}\right). \]

### Step 2: Combine all terms
Now substituting these into the left-hand side (LHS):

\[ \begin{align*} \text{LHS} &= \left(\cot\left(\frac{\pi}{64}\right) - \cot\left(\frac{\pi}{32}\right)\right) + \left(\cot\left(\frac{\pi}{32}\right) - \cot\left(\frac{\pi}{16}\right)\right) + \left(\cot\left(\frac{\pi}{16}\right) - \cot\left(\frac{\pi}{8}\right)\right) \\ &\quad + \left(\cot\left(\frac{\pi}{8}\right) - \cot\left(\frac{\pi}{4}\right)\right) + \cot\left(\frac{\pi}{4}\right). \end{align*} \]

### Step 3: Simplify the expression
Notice that all intermediate cotangent terms cancel out:

\[ \text{LHS} = \cot\left(\frac{\pi}{64}\right) - \cot\left(\frac{\pi}{4}\right) + \cot\left(\frac{\pi}{4}\right) = \cot\left(\frac{\pi}{64}\right). \]

### Step 4: Set LHS equal to RHS
Now we have:

\[ \cot\left(\frac{\pi}{64}\right) = \cot\left(\frac{\pi}{k}\right). \]

### Step 5: Compare and solve for k
From the equality of cotangents, we can conclude:

\[ \frac{\pi}{k} = \frac{\pi}{64} \implies k = 64. \]

### Final Answer
Thus, the value of \( k \) is

\[ \boxed{64}. \]