`(sin 2A)/( 1+ cos 2A) . (cos A)/( 1+ cos A)=`

Options

A. `tan""A/2`
B. `cot ""A/2`
C. `sec ""A/2`
D. `cosec ""A/2`

Answer

To solve the expression \(\frac{\sin 2A}{1 + \cos 2A} \cdot \frac{\cos A}{1 + \cos A}\), we will simplify it step by step.

### Step 1: Use Trigonometric Identities
We know that:
\[ \sin 2A = 2 \sin A \cos A \]
and
\[ \cos 2A = 2 \cos^2 A - 1 \]
Thus,
\[ 1 + \cos 2A = 1 + (2 \cos^2 A - 1) = 2 \cos^2 A \]

### Step 2: Substitute the Identities
Now, substituting these identities into the expression:
\[ \frac{\sin 2A}{1 + \cos 2A} = \frac{2 \sin A \cos A}{2 \cos^2 A} = \frac{\sin A}{\cos A} \]

### Step 3: Simplify the First Part
So, we can simplify the first part:
\[ \frac{\sin 2A}{1 + \cos 2A} = \tan A \]

### Step 4: Simplify the Second Part
Now, we simplify the second part:
\[ \frac{\cos A}{1 + \cos A} \]
This part remains as is for now.

### Step 5: Combine the Results
Now we can combine the results:
\[ \tan A \cdot \frac{\cos A}{1 + \cos A} = \frac{\sin A}{\cos A} \cdot \frac{\cos A}{1 + \cos A} \]

### Step 6: Cancel Out \(\cos A\)
Notice that \(\cos A\) in the numerator and denominator cancels out:
\[ \frac{\sin A}{1 + \cos A} \]

### Final Result
Thus, the final simplified expression is:
\[ \frac{\sin A}{1 + \cos A} \]