`(1+sin 2theta+cos 2theta)/(1+sin2 theta-cos 2 theta)`=

Options

A. `(1)/(2)tan theta`
B. `(1)/(2)cot theta`
C. `tan theta`
D. `cot theta`

Answer

To solve the expression \((1 + \sin 2\theta + \cos 2\theta) / (1 + \sin 2\theta - \cos 2\theta)\), we will use trigonometric identities to simplify it step by step.

### Step 1: Substitute the identities
We know that:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]
Substituting these identities into the expression gives us:
\[ \frac{1 + 2\sin \theta \cos \theta + (\cos^2 \theta - \sin^2 \theta)}{1 + 2\sin \theta \cos \theta - (\cos^2 \theta - \sin^2 \theta)} \]

### Step 2: Simplify the numerator
Now, let's simplify the numerator:
\[ 1 + 2\sin \theta \cos \theta + \cos^2 \theta - \sin^2 \theta \]
We can express \(1\) as \(\sin^2 \theta + \cos^2 \theta\):
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta - \sin^2 \theta \]
This simplifies to:
\[ 2\cos^2 \theta + 2\sin \theta \cos \theta \]

### Step 3: Simplify the denominator
Now, let's simplify the denominator:
\[ 1 + 2\sin \theta \cos \theta - (\cos^2 \theta - \sin^2 \theta) \]
Again, substituting \(1\) as \(\sin^2 \theta + \cos^2 \theta\):
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta - \cos^2 \theta + \sin^2 \theta \]
This simplifies to:
\[ 2\sin^2 \theta + 2\sin \theta \cos \theta \]

### Step 4: Rewrite the expression
Now we can rewrite our expression:
\[ \frac{2\cos^2 \theta + 2\sin \theta \cos \theta}{2\sin^2 \theta + 2\sin \theta \cos \theta} \]

### Step 5: Factor out common terms
We can factor out \(2\) from both the numerator and the denominator:
\[ \frac{2(\cos^2 \theta + \sin \theta \cos \theta)}{2(\sin^2 \theta + \sin \theta \cos \theta)} \]
This simplifies to:
\[ \frac{\cos^2 \theta + \sin \theta \cos \theta}{\sin^2 \theta + \sin \theta \cos \theta} \]

### Step 6: Final simplification
Now, we can express this as:
\[ \frac{\cos^2 \theta + \sin \theta \cos \theta}{\sin^2 \theta + \sin \theta \cos \theta} \]

This is the simplified form of the original expression.

### Final Answer
Thus, the value of the expression \(\frac{1 + \sin 2\theta + \cos 2\theta}{1 + \sin 2\theta - \cos 2\theta}\) is:
\[ \frac{\cos^2 \theta + \sin \theta \cos \theta}{\sin^2 \theta + \sin \theta \cos \theta} \]