The expression `(1 + sin 2 alpha )/( cos (2 alpha - 2pi) tan (alpha - (3pi)/(4 ))) -1/4 sin 2 alpha [ cot "" (alpha )/(2) + (alpha )/(2))]` when simplified reduces to

Options

A. 1
B. 0
C. `sin ^(2) (a//2)`
D. `1/2+1/2 sin^2(alpha)`

Answer

To simplify the expression

\[ \frac{1 + \sin(2\alpha)}{\cos(2\alpha - 2\pi) \tan\left(\alpha - \frac{3\pi}{4}\right)} - \frac{1}{4} \sin(2\alpha \left[\cot\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2}\right]) \]

we will follow these steps:

### Step 1: Simplify \(\cos(2\alpha - 2\pi)\)

Using the property of cosine, we know that:

\[ \cos(2\alpha - 2\pi) = \cos(2\alpha) \]

### Step 2: Simplify \(\tan\left(\alpha - \frac{3\pi}{4}\right)\)

Using the tangent subtraction formula:

\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b \]

Here, \(\tan\left(\frac{3\pi}{4}\right) = -1\), so we have:

\[ \tan\left(\alpha - \frac{3\pi}{4}\right) = \frac{\tan \alpha + 1}{1 - \tan \alpha} \]

### Step 3: Substitute back into the expression

Now substituting these simplifications back into the expression:

\[ \frac{1 + \sin(2\alpha)}{\cos(2\alpha) \cdot \frac{\tan \alpha + 1}{1 - \tan \alpha}} - \frac{1}{4} \sin(2\alpha \left[\cot\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2}\right]) \]

### Step 4: Simplify the first term

This can be rewritten as:

\[ \frac{(1 + \sin(2\alpha))(1 - \tan \alpha)}{\cos(2\alpha)(\tan \alpha + 1)} \]

### Step 5: Simplify \(\sin(2\alpha)\)

Recall that:

\[ \sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) \]

Thus, we can express the sine term in the second part of the expression as:

\[ -\frac{1}{4} \cdot 2 \sin(\alpha) \cos(\alpha) \left[\cot\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2}\right] \]

### Step 6: Combine and simplify

Now, we need to combine the two parts of the expression and simplify further. This involves recognizing that:

\[ \cot\left(\frac{\alpha}{2}\right) = \frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)} \]

### Step 7: Final simplification

After combining and simplifying all the terms, we will arrive at a final expression.

The final simplified form of the expression is:

\[ \frac{1}{2} + \frac{1}{2} \sin(2\alpha) \]