In a triangle ABC , if `tanA=2sin2Cand3cosA=2sinBsinC`, then C=

Options

A. `(pi)/(8)`
B. `(pi)/(6)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Answer

To solve the problem step by step, we will analyze the given equations and use trigonometric identities to find the value of angle C in triangle ABC.

### Step 1: Start with the given equations
We have two equations from the problem:
1. \( \tan A = 2 \sin 2C \)
2. \( 3 \cos A = 2 \sin B \sin C \)

### Step 2: Rewrite the second equation
Using the identity \( 2 \sin B \sin C = \cos(B - C) - \cos(B + C) \), we can rewrite the second equation:
\[ 3 \cos A = \cos(B - C) - \cos(B + C) \]
Since \( A + B + C = \pi \), we can express \( B + C \) as \( \pi - A \):
\[ 3 \cos A = \cos(B - C) + \cos A \]
Rearranging gives us:
\[ 3 \cos A - \cos A = \cos(B - C) \]
\[ 2 \cos A = \cos(B - C) \tag{1} \]

### Step 3: Analyze the first equation
From the first equation \( \tan A = 2 \sin 2C \), we can express \( \sin A \) in terms of \( \cos A \):
\[ \sin A = \tan A \cdot \cos A = 2 \sin 2C \cdot \cos A \]

### Step 4: Use the double angle identity
Using the double angle identity for sine, we have:
\[ \sin 2C = 2 \sin C \cos C \]
Thus:
\[ \sin A = 4 \sin C \cos C \cos A \]

### Step 5: Substitute \( 2 \cos A \)
From equation (1), we know \( 2 \cos A = \cos(B - C) \). Replacing \( 2 \cos A \) in our equation gives:
\[ \sin A = \cos(B - C) \cdot \sin C \cos C \]

### Step 6: Use the sine rule
We know that \( A + B + C = \pi \). Therefore, we can express \( B \) in terms of \( A \) and \( C \):
\[ B = \pi - A - C \]
Substituting this into our sine equation gives:
\[ \sin A = \sin(3C - B) \]
Using the sine identity, we can express this as:
\[ \sin A = \sin(3C - (\pi - A - C)) = \sin(4C + A) \]

### Step 7: Set up the equation
From the sine identity, we can equate:
\[ \sin A = \sin(4C + A) \]
This implies:
\[ A = 4C + A \quad \text{or} \quad A + 4C = \pi \]
Thus:
\[ 4C = \pi - A \]

### Step 8: Substitute back into the triangle sum
Using \( A + B + C = \pi \):
\[ A + (\pi - A - C) + C = \pi \]
This simplifies to:
\[ 4C = \pi \]
Thus:
\[ C = \frac{\pi}{4} \]

### Final Answer
The value of angle C is:
\[ C = \frac{\pi}{4} \]