find the value of expression `(sin2x)/(1+cos2x)` is:

Options

A. tan2x
B. cos2x
C. tanx
D. none of these

Answer

To find the value of the expression \(\frac{\sin 2x}{1 + \cos 2x}\), we can follow these steps:

### Step 1: Use the double angle identities
We know the double angle identities:
- \(\sin 2x = 2 \sin x \cos x\)
- \(\cos 2x = 2 \cos^2 x - 1\)

### Step 2: Substitute the identities into the expression
Substituting these identities into the expression, we have:
\[ \frac{\sin 2x}{1 + \cos 2x} = \frac{2 \sin x \cos x}{1 + (2 \cos^2 x - 1)} \]

### Step 3: Simplify the denominator
Now simplify the denominator:
\[ 1 + (2 \cos^2 x - 1) = 2 \cos^2 x \]
So the expression becomes:
\[ \frac{2 \sin x \cos x}{2 \cos^2 x} \]

### Step 4: Cancel common factors
We can cancel the common factor of 2 in the numerator and denominator:
\[ \frac{2 \sin x \cos x}{2 \cos^2 x} = \frac{\sin x \cos x}{\cos^2 x} \]

### Step 5: Further simplify the expression
Now, we can simplify this further:
\[ \frac{\sin x \cos x}{\cos^2 x} = \frac{\sin x}{\cos x} = \tan x \]

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

---