`sqrt((cottheta+costheta)/(cottheta-costheta))` is equal to :

Options

A. `sec theta - tan theta`
B. `1 + sec theta tan theta`
C. `1 - sec theta tan theta`
D. `sec theta + tan theta`

Answer

To solve the expression \( \sqrt{\frac{\cot \theta + \cos \theta}{\cot \theta - \cos \theta}} \), we can follow these steps:

### Step 1: Rewrite cotangent in terms of sine and cosine
Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute this into the expression:

\[ \sqrt{\frac{\frac{\cos \theta}{\sin \theta} + \cos \theta}{\frac{\cos \theta}{\sin \theta} - \cos \theta}} \]

### Step 2: Simplify the numerator and denominator
The numerator becomes:

\[ \frac{\cos \theta}{\sin \theta} + \cos \theta = \frac{\cos \theta + \cos \theta \sin \theta}{\sin \theta} = \frac{\cos \theta (1 + \sin \theta)}{\sin \theta} \]

The denominator becomes:

\[ \frac{\cos \theta}{\sin \theta} - \cos \theta = \frac{\cos \theta - \cos \theta \sin \theta}{\sin \theta} = \frac{\cos \theta (1 - \sin \theta)}{\sin \theta} \]

### Step 3: Substitute back into the square root
Now substitute the simplified numerator and denominator back into the square root:

\[ \sqrt{\frac{\frac{\cos \theta (1 + \sin \theta)}{\sin \theta}}{\frac{\cos \theta (1 - \sin \theta)}{\sin \theta}}} \]

### Step 4: Cancel out common terms
The \( \sin \theta \) in the numerator and denominator cancels out, and the \( \cos \theta \) also cancels out (assuming \( \cos \theta \neq 0 \)):

\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 5: Rationalize the expression
To simplify further, multiply the numerator and denominator by \( 1 + \sin \theta \):

\[ \sqrt{\frac{(1 + \sin \theta)^2}{(1 - \sin \theta)(1 + \sin \theta)}} = \sqrt{\frac{(1 + \sin \theta)^2}{1 - \sin^2 \theta}} \]

### Step 6: Use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \)
Now, replace \( 1 - \sin^2 \theta \) with \( \cos^2 \theta \):

\[ \sqrt{\frac{(1 + \sin \theta)^2}{\cos^2 \theta}} = \frac{1 + \sin \theta}{\cos \theta} \]

### Step 7: Final expression
This can be rewritten as:

\[ \sec \theta + \tan \theta \]

Thus, the final answer is:

\[ \sec \theta + \tan \theta \]