What is the simplified value of `tan ((theta )/( 2)) + cot ((theta )/( 2))` ?

Options

A. 2 cosec `theta`
B. 2 sec `theta`
C. sin` theta`
D. cosec `theta `

Answer

To simplify the expression \( \tan\left(\frac{\theta}{2}\right) + \cot\left(\frac{\theta}{2}\right) \), we can follow these steps:

### Step 1: Rewrite the tangent and cotangent
We know that:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} \]
\[ \cot\left(\frac{\theta}{2}\right) = \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]
Thus, we can rewrite the expression as:
\[ \tan\left(\frac{\theta}{2}\right) + \cot\left(\frac{\theta}{2}\right) = \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} + \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \( \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \). Therefore, we can combine the fractions:
\[ = \frac{\sin^2\left(\frac{\theta}{2}\right) + \cos^2\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)} \]

### Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \sin^2\left(\frac{\theta}{2}\right) + \cos^2\left(\frac{\theta}{2}\right) = 1 \]
Thus, we can simplify the numerator:
\[ = \frac{1}{\sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)} \]

### Step 4: Simplify further using double angle identity
We can use the double angle identity for sine:
\[ \sin\left(\theta\right) = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \]
So, we can rewrite the expression as:
\[ = \frac{2}{\sin\left(\theta\right)} \]

### Step 5: Final result
Thus, the simplified value of \( \tan\left(\frac{\theta}{2}\right) + \cot\left(\frac{\theta}{2}\right) \) is:
\[ = 2 \cdot \frac{1}{\sin\left(\theta\right)} = \frac{2}{\sin\left(\theta\right)} \]