`tan^2 (pi/16) + tan^2 ((2pi)/16) + tan^2((3pi)/16).........+tan^2((7pi)/16/)=35`

Options

A. 24
B. 34
C. 44
D. None of these

Answer

To solve the problem \( \tan^2 \left( \frac{\pi}{16} \right) + \tan^2 \left( \frac{2\pi}{16} \right) + \tan^2 \left( \frac{3\pi}{16} \right) + \tan^2 \left( \frac{4\pi}{16} \right) + \tan^2 \left( \frac{5\pi}{16} \right) + \tan^2 \left( \frac{6\pi}{16} \right) + \tan^2 \left( \frac{7\pi}{16} \right) \), we can use a known formula for the sum of squares of tangents.

### Step 1: Identify the formula
The formula for the sum of squares of tangents is given by:
\[ \sum_{k=1}^{n-1} \tan^2 \left( \frac{k \pi}{2n} \right) = \frac{(n-1)(2n-1)}{3} \]

### Step 2: Determine \(n\)
In our case, we can see that we are summing from \(k=1\) to \(k=7\) (which is \(n-1\)), and the angles are in the form \( \frac{k \pi}{16} \). This means \(2n = 16\) or \(n = 8\).

### Step 3: Substitute \(n\) into the formula
Now, substituting \(n = 8\) into the formula:
\[ \sum_{k=1}^{7} \tan^2 \left( \frac{k \pi}{16} \right) = \frac{(8-1)(2 \cdot 8 - 1)}{3} \]

### Step 4: Simplify the expression
Calculating the values:
\[ = \frac{(7)(15)}{3} \]
\[ = \frac{105}{3} = 35 \]

### Conclusion
Thus, the sum \( \tan^2 \left( \frac{\pi}{16} \right) + \tan^2 \left( \frac{2\pi}{16} \right) + \tan^2 \left( \frac{3\pi}{16} \right) + \tan^2 \left( \frac{4\pi}{16} \right) + \tan^2 \left( \frac{5\pi}{16} \right) + \tan^2 \left( \frac{6\pi}{16} \right) + \tan^2 \left( \frac{7\pi}{16} \right) = 35\).