`(sin theta[(1-tantheta)tantheta+sec^(2)theta])/((1-sin theta)tantheta(1+tantheta)(sec theta+tantheta))` is equal to :

Options

A. `sin thetacostheta`
B. `-1`
C. `"cosec"thetasectheta`
D. 1

Answer

To solve the expression

\[ \frac{\sin \theta \left[ (1 - \tan \theta) \tan \theta + \sec^2 \theta \right]}{(1 - \sin \theta) \tan \theta (1 + \tan \theta)(\sec \theta + \tan \theta)} \]

we will simplify it step by step.

### Step 1: Rewrite the expression
Start by rewriting the expression clearly:
\[ \frac{\sin \theta \left[ (1 - \tan \theta) \tan \theta + \sec^2 \theta \right]}{(1 - \sin \theta) \tan \theta (1 + \tan \theta)(\sec \theta + \tan \theta)} \]

### Step 2: Expand the numerator
Expand the numerator:
\[ (1 - \tan \theta) \tan \theta + \sec^2 \theta = \tan \theta - \tan^2 \theta + \sec^2 \theta \]
Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), we can substitute:
\[ \tan \theta - \tan^2 \theta + (1 + \tan^2 \theta) = \tan \theta + 1 \]
Thus, the numerator becomes:
\[ \sin \theta (\tan \theta + 1) \]

### Step 3: Rewrite the denominator
Now, let's rewrite the denominator:
\[ (1 - \sin \theta) \tan \theta (1 + \tan \theta)(\sec \theta + \tan \theta) \]
We know \(\sec \theta = \frac{1}{\cos \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so:
\[ \sec \theta + \tan \theta = \frac{1 + \sin \theta}{\cos \theta} \]
Substituting this into the denominator gives:
\[ (1 - \sin \theta) \tan \theta (1 + \tan \theta) \cdot \frac{1 + \sin \theta}{\cos \theta} \]

### Step 4: Substitute \(\tan \theta\)
Substituting \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) into the denominator:
\[ (1 - \sin \theta) \cdot \frac{\sin \theta}{\cos \theta} \cdot \left(1 + \frac{\sin \theta}{\cos \theta}\right) \cdot \frac{1 + \sin \theta}{\cos \theta} \]
This simplifies to:
\[ (1 - \sin \theta) \cdot \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta + \sin \theta}{\cos \theta} \cdot \frac{1 + \sin \theta}{\cos \theta} \]
Thus, the denominator becomes:
\[ (1 - \sin \theta) \cdot \frac{\sin \theta (1 + \sin \theta)(\cos + \sin)}{\cos^3 \theta} \]

### Step 5: Simplify the entire expression
Now we can combine the numerator and denominator:
\[ \frac{\sin \theta (\tan \theta + 1)}{(1 - \sin \theta) \cdot \frac{\sin \theta (1 + \sin \theta)(\cos + \sin)}{\cos^3 \theta}} \]
Cancelling \(\sin \theta\) from the numerator and denominator gives:
\[ \frac{\tan \theta + 1}{(1 - \sin \theta) \cdot \frac{(1 + \sin \theta)(\cos + \sin)}{\cos^3 \theta}} \]

### Step 6: Final simplification
The expression simplifies down to:
\[ \frac{\tan \theta + 1}{(1 - \sin \theta)(1 + \sin \theta)} \cdot \cos^3 \theta \]
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\), we have:
\[ \frac{\tan \theta + 1}{\cos^2 \theta} \cdot \cos^3 \theta = \tan \theta + 1 \]

### Final Result
Thus, the final simplified expression is:
\[ 1 \]