The value of ` tan 5 theta` is

Options

A. `(5 tantheta-10tan^(3)theta+tan^(5)theta)/(1-10tan^(2)theta+5 tan^(4)theta)`
B. `(5 tantheta+10tan^(3)theta-tan^(5)theta)/(1+10tan^(2)theta-5 tan^(4)theta)`
C. `(5tan^(5)theta-10tan ^(3)+tantheta)/(1-10 tan^(2)theta+5tan^(4)theta)`
D. none of these

Answer

To find the value of \( \tan 5\theta \), we can use the angle addition formula for tangent and the known formulas for \( \tan 2\theta \) and \( \tan 3\theta \).

### Step-by-Step Solution:

1. **Express \( \tan 5\theta \)**:
\[ \tan 5\theta = \tan(3\theta + 2\theta) \]

2. **Use the tangent addition formula**:
The formula for \( \tan(x + y) \) is:
\[ \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \]
Applying this to our expression:
\[ \tan 5\theta = \frac{\tan 3\theta + \tan 2\theta}{1 - \tan 3\theta \tan 2\theta} \]

3. **Substitute \( \tan 3\theta \) and \( \tan 2\theta \)**:
- The formula for \( \tan 3\theta \) is:
\[ \tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} \]
- The formula for \( \tan 2\theta \) is:
\[ \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \]

4. **Substituting these into the expression for \( \tan 5\theta \)**:
\[ \tan 5\theta = \frac{\frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} + \frac{2\tan\theta}{1 - \tan^2\theta}}{1 - \left(\frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}\right)\left(\frac{2\tan\theta}{1 - \tan^2\theta}\right)} \]

5. **Finding a common denominator for the numerator**:
The common denominator for the terms in the numerator is:
\[ (1 - 3\tan^2\theta)(1 - \tan^2\theta) \]
Thus, we rewrite the numerator:
\[ \tan 5\theta = \frac{(3\tan\theta - \tan^3\theta)(1 - \tan^2\theta) + 2\tan\theta(1 - 3\tan^2\theta)}{(1 - 3\tan^2\theta)(1 - \tan^2\theta)} \]

6. **Simplifying the numerator**:
Expanding the numerator:
\[ (3\tan\theta - \tan^3\theta)(1 - \tan^2\theta) + 2\tan\theta(1 - 3\tan^2\theta) \]
This results in:
\[ 3\tan\theta - 3\tan^3\theta - \tan^3\theta + \tan^5\theta + 2\tan\theta - 6\tan^3\theta \]
Combining like terms gives:
\[ 5\tan\theta - 10\tan^3\theta + \tan^5\theta \]

7. **Simplifying the denominator**:
The denominator simplifies to:
\[ (1 - 3\tan^2\theta)(1 - \tan^2\theta) = 1 - 4\tan^2\theta + 3\tan^4\theta \]

8. **Final expression for \( \tan 5\theta \)**:
Thus, we have:
\[ \tan 5\theta = \frac{5\tan\theta - 10\tan^3\theta + \tan^5\theta}{1 - 4\tan^2\theta + 3\tan^4\theta} \]