`(1+costheta)/(sintheta)=?`

Options

A. `tan""(theta)/(2)`
B. `cot""(theta)/(2)`
C. `tantheta`
D. `cottheta`

Answer

To solve the expression \((1 + \cos \theta) / \sin \theta\), we can follow these steps:

### Step-by-Step Solution

1. **Start with the expression**:
\[ \frac{1 + \cos \theta}{\sin \theta} \]

2. **Use the identity for \(\cos \theta\)**:
We can rewrite \(\cos \theta\) using the double angle identity:
\[ \cos \theta = 2 \cos^2 \left(\frac{\theta}{2}\right) - 1 \]
Therefore, we can substitute this into our expression:
\[ 1 + \cos \theta = 1 + (2 \cos^2 \left(\frac{\theta}{2}\right) - 1) = 2 \cos^2 \left(\frac{\theta}{2}\right) \]

3. **Substitute back into the expression**:
Now, substituting this back into our original expression gives:
\[ \frac{2 \cos^2 \left(\frac{\theta}{2}\right)}{\sin \theta} \]

4. **Use the identity for \(\sin \theta\)**:
We can also express \(\sin \theta\) using the double angle identity:
\[ \sin \theta = 2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right) \]

5. **Substitute \(\sin \theta\) into the expression**:
Now substituting this into our expression gives:
\[ \frac{2 \cos^2 \left(\frac{\theta}{2}\right)}{2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)} \]

6. **Simplify the expression**:
The \(2\)s in the numerator and denominator cancel out:
\[ \frac{\cos^2 \left(\frac{\theta}{2}\right)}{\sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)} \]
This can be simplified further:
\[ \frac{\cos \left(\frac{\theta}{2}\right)}{\sin \left(\frac{\theta}{2}\right)} = \cot \left(\frac{\theta}{2}\right) \]

7. **Final result**:
Thus, we conclude that:
\[ \frac{1 + \cos \theta}{\sin \theta} = \cot \left(\frac{\theta}{2}\right) \]