Find the value of `sqrt((1-sintheta)/(1+sintheta))`.

Options

A. `sectheta-tantheta`
B. `costheta-cottheta`
C. `sintheta-cosectheta`
D. `cosectheta-cottheta`

Answer

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

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

### Step 2: Multiply numerator and denominator by \(1 - \sin \theta\)
To simplify the expression, we can multiply both 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 numerator and denominator
The numerator becomes:
\[ (1 - \sin \theta)^2 \]
The denominator can be simplified using the difference of squares:
\[ (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta \]
Thus, we have:
\[ \sqrt{\frac{(1 - \sin \theta)^2}{\cos^2 \theta}} \]

### Step 4: Simplify the square root
Taking the square root of the fraction gives:
\[ \frac{1 - \sin \theta}{\cos \theta} \]

### Step 5: Rewrite in terms of trigonometric functions
The expression can be rewritten using trigonometric identities:
\[ \frac{1 - \sin \theta}{\cos \theta} = \sec \theta - \tan \theta \]

### Final Result
Thus, the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \) is:
\[ \sec \theta - \tan \theta \]

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