The value of `(sin(pi-alpha))/(sin alpha-cos alpha tan.(alpha)/(2))-cos alpha` is

Options

A. `-2`
B. `-1`
C. 1
D. 2

Explanation

To solve the expression \(\frac{\sin(\pi - \alpha)}{\sin \alpha - \cos \alpha \cdot \tan\left(\frac{\alpha}{2}\right)} - \cos \alpha\), we will follow these steps:

### Step 1: Simplify \(\sin(\pi - \alpha)\)
Using the identity \(\sin(\pi - \theta) = \sin \theta\), we can simplify:
\[ \sin(\pi - \alpha) = \sin \alpha \]
Thus, the expression becomes:
\[ \frac{\sin \alpha}{\sin \alpha - \cos \alpha \cdot \tan\left(\frac{\alpha}{2}\right)} - \cos \alpha \]

**Hint:** Remember the sine subtraction identity to simplify \(\sin(\pi - \alpha)\).

### Step 2: Rewrite \(\tan\left(\frac{\alpha}{2}\right)\)
We know that:
\[ \tan\left(\frac{\alpha}{2}\right) = \frac{\sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)} \]
Substituting this into the expression gives:
\[ \frac{\sin \alpha}{\sin \alpha - \cos \alpha \cdot \frac{\sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)}} - \cos \alpha \]

**Hint:** Use the definition of tangent in terms of sine and cosine.

### Step 3: Combine the terms in the denominator
The denominator now becomes:
\[ \sin \alpha - \frac{\cos \alpha \cdot \sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)} \]
To combine, we can take a common denominator:
\[ \frac{\sin \alpha \cdot \cos\left(\frac{\alpha}{2}\right) - \cos \alpha \cdot \sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)} \]

**Hint:** Make sure to factor out the common denominator correctly.

### Step 4: Substitute back into the expression
Now the expression becomes:
\[ \frac{\sin \alpha \cdot \cos\left(\frac{\alpha}{2}\right)}{\sin \alpha \cdot \cos\left(\frac{\alpha}{2}\right) - \cos \alpha \cdot \sin\left(\frac{\alpha}{2}\right)} - \cos \alpha \]

**Hint:** Keep track of the numerator and denominator when substituting.

### Step 5: Simplify using identities
Using the identity \(\sin \alpha = 2 \sin\left(\frac{\alpha}{2}\right) \cos\left(\frac{\alpha}{2}\right)\), we can rewrite:
\[ \frac{2 \sin\left(\frac{\alpha}{2}\right) \cos^2\left(\frac{\alpha}{2}\right)}{2 \sin\left(\frac{\alpha}{2}\right) \cos^2\left(\frac{\alpha}{2}\right) - \cos \alpha \cdot \sin\left(\frac{\alpha}{2}\right)} \]

**Hint:** Use double angle formulas to express sine and cosine in terms of half angles.

### Step 6: Factor and simplify
The expression simplifies further:
\[ \frac{2 \cos^2\left(\frac{\alpha}{2}\right)}{2 \cos^2\left(\frac{\alpha}{2}\right) - \cos \alpha} \]
Now, using the identity \( \cos \alpha = 2 \cos^2\left(\frac{\alpha}{2}\right) - 1 \):
\[ = \frac{2 \cos^2\left(\frac{\alpha}{2}\right)}{1 + \cos \alpha} \]

**Hint:** Remember the relationship between cosine and half angles.

### Step 7: Final simplification
After substituting and simplifying, we find that the expression evaluates to:
\[ 1 \]

Thus, the final answer is:
\[ \boxed{1} \]