What is `sqrt((1 + sin theta)/(1 -sintheta))` equal to ?

Options

A. `sec theta - tan theta`
B. `sec theta + tan theta`
C. `cosec theta + cot theta`
D. `cosec theta -cot theta`

Answer

To solve the expression \( \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 2: Rationalize the expression
To simplify this expression, we can rationalize it by multiplying the numerator and the denominator by \( 1 + \sin \theta \):
\[ \sqrt{\frac{(1 + \sin \theta)(1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)}} \]

### Step 3: Simplify the denominator
Using the difference of squares in the denominator:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]
Thus, we can rewrite the expression as:
\[ \sqrt{\frac{(1 + \sin \theta)^2}{\cos^2 \theta}} \]

### Step 4: Simplify the square root
Now, we can take the square root of the numerator and the denominator:
\[ \frac{1 + \sin \theta}{\cos \theta} \]

### Step 5: Rewrite using trigonometric identities
We can express this in terms of trigonometric functions:
\[ \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \sec \theta + \tan \theta \]

### Final Answer
Thus, the expression simplifies to:
\[ \sec \theta + \tan \theta \]

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