If `tan((pi)/(2)-(theta)/(2))=sqrt(3)`, then the value of `costheta` is.

Options

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

Answer

To solve the problem where \( \tan\left(\frac{\pi}{2} - \frac{\theta}{2}\right) = \sqrt{3} \), we can follow these steps:

### Step 1: Use the cotangent identity
We know that:
\[ \tan\left(\frac{\pi}{2} - x\right) = \cot(x) \]
Thus, we can rewrite the equation as:
\[ \cot\left(\frac{\theta}{2}\right) = \sqrt{3} \]

### Step 2: Relate cotangent to angles
The cotangent of an angle is equal to \(\sqrt{3}\) when the angle is \(30^\circ\) (or \(\frac{\pi}{6}\) radians). Therefore, we can write:
\[ \frac{\theta}{2} = 30^\circ \]

### Step 3: Solve for \(\theta\)
To find \(\theta\), we multiply both sides by 2:
\[ \theta = 2 \times 30^\circ = 60^\circ \]

### Step 4: Find \(\cos(\theta)\)
Now, we need to find \(\cos(\theta)\):
\[ \cos(60^\circ) = \frac{1}{2} \]

### Conclusion
Thus, the value of \(\cos(\theta)\) is:
\[ \cos(\theta) = \frac{1}{2} \]