If `tan((pi)/(2)-(theta)/(2))=sqrt(3)` , the value of `sectheta` is :

Options

A. 0
B. `(1)/(sqrt(2))`
C. `2`
D. 1

Answer

To solve the problem, we start with the given equation:

1. **Given**:
\[ \tan\left(\frac{\pi}{2} - \frac{\theta}{2}\right) = \sqrt{3} \]

2. **Using the co-function identity**:
The tangent of an angle can be transformed using the co-function identity:
\[ \tan\left(\frac{\pi}{2} - x\right) = \cot(x) \]
Therefore, we can rewrite the equation as:
\[ \cot\left(\frac{\theta}{2}\right) = \sqrt{3} \]

3. **Finding the angle**:
The cotangent of an angle is equal to \(\sqrt{3}\) when the angle is:
\[ \frac{\theta}{2} = \frac{\pi}{6} \quad \text{(since } \cot\left(\frac{\pi}{6}\right) = \sqrt{3}\text{)} \]

4. **Solving for \(\theta\)**:
To find \(\theta\), we multiply both sides by 2:
\[ \theta = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

5. **Finding \(\sec(\theta)\)**:
Now that we have \(\theta\), we can find \(\sec(\theta)\):
\[ \sec(\theta) = \sec\left(\frac{\pi}{3}\right) \]
The secant function is the reciprocal of the cosine function:
\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} \]
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Therefore:
\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} = 2 \]

6. **Final answer**:
Thus, the value of \(\sec(\theta)\) is:
\[ \sec(\theta) = 2 \]