The number of values of x for which `sin2x + cos4x = 2` is

Options

A. Zero
B. 1
C. 2
D. Infinite

Answer

To solve the equation \( \sin 2x + \cos 4x = 2 \), we can follow these steps:

### Step 1: Analyze the equation
The maximum value of \( \sin 2x \) is 1 and the maximum value of \( \cos 4x \) is also 1. Therefore, the maximum value of \( \sin 2x + \cos 4x \) can be at most \( 1 + 1 = 2 \).

### Step 2: Determine when the maximum is achieved
For \( \sin 2x + \cos 4x = 2 \) to hold true, both \( \sin 2x \) and \( \cos 4x \) must simultaneously equal their maximum values:
- \( \sin 2x = 1 \)
- \( \cos 4x = 1 \)

### Step 3: Solve for \( \sin 2x = 1 \)
The equation \( \sin 2x = 1 \) occurs when:
\[ 2x = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]
This simplifies to:
\[ x = \frac{\pi}{4} + n\pi \]

### Step 4: Solve for \( \cos 4x = 1 \)
The equation \( \cos 4x = 1 \) occurs when:
\[ 4x = 2m\pi \quad (m \in \mathbb{Z}) \]
This simplifies to:
\[ x = \frac{m\pi}{2} \]

### Step 5: Find common solutions
We need to find the values of \( x \) that satisfy both conditions:
1. \( x = \frac{\pi}{4} + n\pi \)
2. \( x = \frac{m\pi}{2} \)

Setting these equal gives:
\[ \frac{\pi}{4} + n\pi = \frac{m\pi}{2} \]
Multiplying through by 4 to eliminate the fractions:
\[ \pi + 4n\pi = 2m\pi \]
This simplifies to:
\[ 1 + 4n = 2m \]
Rearranging gives:
\[ 2m - 4n = 1 \]
This is a linear Diophantine equation in \( m \) and \( n \).

### Step 6: Analyze the solutions
The equation \( 2m - 4n = 1 \) has integer solutions. Since \( n \) can take any integer value, we can find corresponding integer values for \( m \). Thus, there are infinitely many pairs \( (m, n) \) that satisfy this equation.

### Conclusion
Since there are infinitely many integer pairs \( (m, n) \) that satisfy the equations derived from \( \sin 2x = 1 \) and \( \cos 4x = 1 \), the number of values of \( x \) for which \( \sin 2x + \cos 4x = 2 \) is infinite.

### Final Answer
The number of values of \( x \) for which \( \sin 2x + \cos 4x = 2 \) is **infinite**.

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