If `sec theta =1 (1)/(4),` then `tan ""(theta)/(2)=`

Options

A. `1/3`
B. `3/4`
C. `1/4`
D. `5/4`

Answer

To solve the problem, we need to find the value of \( \tan \left( \frac{\theta}{2} \right) \) given that \( \sec \theta = 1 \frac{1}{4} \).

### Step-by-Step Solution:

1. **Convert Secant to Improper Fraction**:
\[ \sec \theta = 1 \frac{1}{4} = \frac{5}{4} \]

**Hint**: Remember that a mixed number can be converted to an improper fraction by multiplying the whole number by the denominator and adding the numerator.

2. **Find Cosine**:
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can find \( \cos \theta \) as follows:
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]

**Hint**: The secant function is the reciprocal of the cosine function.

3. **Use Half Angle Identity for Cosine**:
The half-angle identity for cosine is:
\[ \cos \theta = \frac{1 - \tan^2 \left( \frac{\theta}{2} \right)}{1 + \tan^2 \left( \frac{\theta}{2} \right)} \]
Plugging in our value for \( \cos \theta \):
\[ \frac{4}{5} = \frac{1 - \tan^2 \left( \frac{\theta}{2} \right)}{1 + \tan^2 \left( \frac{\theta}{2} \right)} \]

**Hint**: This identity relates the cosine of an angle to the tangent of half that angle.

4. **Cross Multiply**:
Cross multiplying gives us:
\[ 4(1 + \tan^2 \left( \frac{\theta}{2} \right)) = 5(1 - \tan^2 \left( \frac{\theta}{2} \right)) \]
Expanding both sides:
\[ 4 + 4\tan^2 \left( \frac{\theta}{2} \right) = 5 - 5\tan^2 \left( \frac{\theta}{2} \right) \]

**Hint**: When cross multiplying, ensure you distribute correctly.

5. **Rearranging the Equation**:
Rearranging the equation to isolate terms involving \( \tan^2 \left( \frac{\theta}{2} \right) \):
\[ 4 + 4\tan^2 \left( \frac{\theta}{2} \right) + 5\tan^2 \left( \frac{\theta}{2} \right) = 5 \]
This simplifies to:
\[ 9\tan^2 \left( \frac{\theta}{2} \right) = 5 - 4 \]
\[ 9\tan^2 \left( \frac{\theta}{2} \right) = 1 \]

**Hint**: Combine like terms carefully to simplify the equation.

6. **Solve for \( \tan^2 \left( \frac{\theta}{2} \right) \)**:
Dividing both sides by 9:
\[ \tan^2 \left( \frac{\theta}{2} \right) = \frac{1}{9} \]

**Hint**: To isolate \( \tan^2 \), divide both sides by the coefficient in front.

7. **Find \( \tan \left( \frac{\theta}{2} \right) \)**:
Taking the square root of both sides gives:
\[ \tan \left( \frac{\theta}{2} \right) = \frac{1}{3} \]

**Hint**: Remember that taking the square root can yield both positive and negative values, but in this context, we take the positive root since angles in trigonometry are often considered in the first quadrant.

### Final Answer:
\[ \tan \left( \frac{\theta}{2} \right) = \frac{1}{3} \]