What is `cottheta+ cosectheta`equal to?

Options

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

Answer

To solve the problem of finding what \( \cot \theta + \csc \theta \) is equal to, we can follow these steps:

### Step 1: Write the definitions of cotangent and cosecant
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
\[ \csc \theta = \frac{1}{\sin \theta} \]

### Step 2: Combine the two fractions
Now, we can express \( \cot \theta + \csc \theta \) as:
\[ \cot \theta + \csc \theta = \frac{\cos \theta}{\sin \theta} + \frac{1}{\sin \theta} \]
Since both terms have the same denominator, we can combine them:
\[ \cot \theta + \csc \theta = \frac{\cos \theta + 1}{\sin \theta} \]

### Step 3: Simplify the numerator
Next, we can use the identity \( 1 + \cos \theta = 2 \sin^2 \left( \frac{\theta}{2} \right) \):
\[ \cot \theta + \csc \theta = \frac{2 \sin^2 \left( \frac{\theta}{2} \right)}{\sin \theta} \]

### Step 4: Use the double angle identity for sine
We know that \( \sin \theta = 2 \sin \left( \frac{\theta}{2} \right) \cos \left( \frac{\theta}{2} \right) \). Substituting this into our expression gives:
\[ \cot \theta + \csc \theta = \frac{2 \sin^2 \left( \frac{\theta}{2} \right)}{2 \sin \left( \frac{\theta}{2} \right) \cos \left( \frac{\theta}{2} \right)} \]

### Step 5: Cancel out common factors
The \( 2 \) in the numerator and denominator cancels out, and we can also cancel one \( \sin \left( \frac{\theta}{2} \right) \):
\[ \cot \theta + \csc \theta = \frac{\sin \left( \frac{\theta}{2} \right)}{\cos \left( \frac{\theta}{2} \right)} \]

### Step 6: Final simplification
The expression \( \frac{\sin \left( \frac{\theta}{2} \right)}{\cos \left( \frac{\theta}{2} \right)} \) is equal to \( \tan \left( \frac{\theta}{2} \right) \):
\[ \cot \theta + \csc \theta = \tan \left( \frac{\theta}{2} \right) \]

### Conclusion
Thus, we conclude that:
\[ \cot \theta + \csc \theta = \tan \left( \frac{\theta}{2} \right) \]

---