Find the value of `tantheta(1+sec2theta)(1+sec4theta)(1+sec8theta)` .

Options

A. `tan10theta`
B. `tan8theta`
C. `tan12theta`
D. 1

Answer

To solve the problem, we need to find the value of the expression \( \tan \theta (1 + \sec^2 \theta)(1 + \sec^4 \theta)(1 + \sec^8 \theta) \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ \tan \theta (1 + \sec^2 \theta)(1 + \sec^4 \theta)(1 + \sec^8 \theta) \]
Recall that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), so we can rewrite each term:
\[ 1 + \sec^2 \theta = 1 + \frac{1}{\cos^2 \theta} = \frac{\cos^2 \theta + 1}{\cos^2 \theta} \]
Similarly,
\[ 1 + \sec^4 \theta = 1 + \frac{1}{\cos^4 \theta} = \frac{\cos^4 \theta + 1}{\cos^4 \theta} \]
and
\[ 1 + \sec^8 \theta = 1 + \frac{1}{\cos^8 \theta} = \frac{\cos^8 \theta + 1}{\cos^8 \theta} \]

2. **Combine the Terms**:
Now we can express the entire product:
\[ \tan \theta \cdot \frac{\cos^2 \theta + 1}{\cos^2 \theta} \cdot \frac{\cos^4 \theta + 1}{\cos^4 \theta} \cdot \frac{\cos^8 \theta + 1}{\cos^8 \theta} \]
This simplifies to:
\[ \tan \theta \cdot \frac{(\cos^2 \theta + 1)(\cos^4 \theta + 1)(\cos^8 \theta + 1)}{\cos^2 \theta \cos^4 \theta \cos^8 \theta} \]

3. **Substituting \( \tan \theta \)**:
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \):
\[ \frac{\sin \theta}{\cos \theta} \cdot \frac{(\cos^2 \theta + 1)(\cos^4 \theta + 1)(\cos^8 \theta + 1)}{\cos^2 \theta \cos^4 \theta \cos^8 \theta} \]

4. **Using Trigonometric Identities**:
We can use the identity \( 1 + \cos x = 2 \cos^2 \left(\frac{x}{2}\right) \):
- \( \cos^2 \theta + 1 = 2 \cos^2 \left(\frac{\theta}{2}\right) \)
- \( \cos^4 \theta + 1 = 2 \cos^4 \left(\frac{\theta}{2}\right) \)
- \( \cos^8 \theta + 1 = 2 \cos^8 \left(\frac{\theta}{2}\right) \)

5. **Final Simplification**:
Plugging these back into the expression, we can simplify:
\[ \tan \theta \cdot \frac{2 \cos^2 \left(\frac{\theta}{2}\right) \cdot 2 \cos^4 \left(\frac{\theta}{2}\right) \cdot 2 \cos^8 \left(\frac{\theta}{2}\right)}{\cos^2 \theta \cos^4 \theta \cos^8 \theta} \]
This leads to:
\[ \tan \theta \cdot 8 \cdot \frac{\cos^{14} \left(\frac{\theta}{2}\right)}{\cos^{14} \theta} \]
Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express this as:
\[ 8 \cdot \frac{\sin \theta \cos^{14} \left(\frac{\theta}{2}\right)}{\cos^{15} \theta} \]

6. **Final Result**:
After simplification, we find that the value of the expression is:
\[ \tan(8\theta) \]

### Conclusion:
Thus, the value of \( \tan \theta (1 + \sec^2 \theta)(1 + \sec^4 \theta)(1 + \sec^8 \theta) \) is \( \tan(8\theta) \).