`(sin2A)/(1+cos 2A)` is

Options

A. `tan2A`
B. `cos 2A`
C. `tanA`
D. none of these

Answer

To solve the expression \(\frac{\sin 2A}{1 + \cos 2A}\), we can use trigonometric identities. Here’s a step-by-step solution:

### Step 1: Use the double angle identity for sine
We know that:
\[ \sin 2A = 2 \sin A \cos A \]
So we can rewrite the numerator:
\[ \sin 2A = 2 \sin A \cos A \]

### Step 2: Use the double angle identity for cosine
We also know that:
\[ \cos 2A = 2 \cos^2 A - 1 \]
Thus, we can express \(1 + \cos 2A\) as:
\[ 1 + \cos 2A = 1 + (2 \cos^2 A - 1) = 2 \cos^2 A \]

### Step 3: Substitute the identities into the expression
Now we can substitute these identities into our original expression:
\[ \frac{\sin 2A}{1 + \cos 2A} = \frac{2 \sin A \cos A}{2 \cos^2 A} \]

### Step 4: Simplify the expression
We can simplify the fraction:
\[ \frac{2 \sin A \cos A}{2 \cos^2 A} = \frac{\sin A}{\cos A} \]

### Step 5: Recognize the final expression
The expression \(\frac{\sin A}{\cos A}\) is the definition of the tangent function:
\[ \frac{\sin A}{\cos A} = \tan A \]

### Final Result
Thus, we conclude that:
\[ \frac{\sin 2A}{1 + \cos 2A} = \tan A \]

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