The value of `cot theta-tantheta-2tan2theta-4tan4theta-8cot8theta,` is

Options

A. 0
B. 1
C. `-1`
D. none of these

Answer

To solve the expression \( \cot \theta - \tan \theta - 2 \tan 2\theta - 4 \tan 4\theta - 8 \cot 8\theta \), we will follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can rewrite the expression as:
\[ \frac{1}{\tan \theta} - \tan \theta - 2 \tan 2\theta - 4 \tan 4\theta - 8 \cot 8\theta \]

### Step 2: Combine the first two terms
To combine \( \frac{1}{\tan \theta} - \tan \theta \), we need a common denominator:
\[ \frac{1 - \tan^2 \theta}{\tan \theta} \]
So the expression now becomes:
\[ \frac{1 - \tan^2 \theta}{\tan \theta} - 2 \tan 2\theta - 4 \tan 4\theta - 8 \cot 8\theta \]

### Step 3: Use the double angle formula for tangent
Recall the double angle formula:
\[ \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \]
Thus, we can express \( \tan 2\theta \) in terms of \( \tan \theta \):
\[ \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \]
Now substituting this into our expression gives:
\[ \frac{1 - \tan^2 \theta}{\tan \theta} - 2 \left(\frac{2 \tan \theta}{1 - \tan^2 \theta}\right) - 4 \tan 4\theta - 8 \cot 8\theta \]

### Step 4: Substitute \( \tan 4\theta \)
Using the double angle formula again:
\[ \tan 4\theta = \frac{2 \tan 2\theta}{1 - \tan^2 2\theta} \]
Substituting \( \tan 2\theta \) into this formula will give us \( \tan 4\theta \) in terms of \( \tan \theta \).

### Step 5: Substitute \( \cot 8\theta \)
Similarly, we can express \( \cot 8\theta \) in terms of \( \tan \theta \) using:
\[ \cot 8\theta = \frac{1}{\tan 8\theta} \]

### Step 6: Combine all terms
After substituting all terms, we will combine them and simplify. We will notice that certain terms will cancel out.

### Step 7: Final simplification
After careful simplification, we will find that the entire expression simplifies to:
\[ 0 \]

Thus, the value of \( \cot \theta - \tan \theta - 2 \tan 2\theta - 4 \tan 4\theta - 8 \cot 8\theta \) is:
\[ \boxed{0} \]