`((1+tantheta+sectheta)(1+cottheta-cosectheta))/((sectheta+tantheta)(1-sintheta))` is equal to:

Options

A. `2cosectheta`
B. `cosectheta`
C. `sectheta`
D. `2sectheta`

Answer

To solve the expression \(\frac{(1+\tan\theta+\sec\theta)(1+\cot\theta-\csc\theta)}{(\sec\theta+\tan\theta)(1-\sin\theta)}\), we will simplify it step by step.

### Step 1: Substitute Trigonometric Values
We will substitute \(\theta = 30^\circ\) into the expression. The trigonometric values for \(30^\circ\) are:
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
- \(\sec 30^\circ = \frac{2}{\sqrt{3}}\)
- \(\cot 30^\circ = \sqrt{3}\)
- \(\csc 30^\circ = 2\)
- \(\sin 30^\circ = \frac{1}{2}\)

### Step 2: Substitute Values into the Expression
Now we substitute these values into the expression:

\[ 1 + \tan 30^\circ + \sec 30^\circ = 1 + \frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}} = 1 + \frac{3}{\sqrt{3}} = 1 + \sqrt{3} \]

\[ 1 + \cot 30^\circ - \csc 30^\circ = 1 + \sqrt{3} - 2 = \sqrt{3} - 1 \]

\[ \sec 30^\circ + \tan 30^\circ = \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

\[ 1 - \sin 30^\circ = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 3: Substitute into the Main Expression
Now substituting these results into the main expression:

\[ \frac{(1 + \sqrt{3})(\sqrt{3} - 1)}{(\sqrt{3})(\frac{1}{2})} \]

### Step 4: Simplify the Numerator
Now we simplify the numerator:

\[ (1 + \sqrt{3})(\sqrt{3} - 1) = 1 \cdot \sqrt{3} - 1 + \sqrt{3} \cdot \sqrt{3} - \sqrt{3} = \sqrt{3} - 1 + 3 - \sqrt{3} = 2 \]

### Step 5: Simplify the Denominator
Now simplifying the denominator:

\[ \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \]

### Step 6: Final Simplification
Now we can simplify the entire expression:

\[ \frac{2}{\frac{\sqrt{3}}{2}} = 2 \cdot \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Step 7: Rationalize the Denominator
To rationalize the denominator:

\[ \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]

### Final Answer
Thus, the final answer is:

\[ \frac{4\sqrt{3}}{3} \]