If x and y are positive acute angles such that sin(4x - y) = 1 and cos(2x + y) = 1/2, then what is the value of cot (x + 2y)?

Options

A. `sqrt3`
B. `1//sqrt3`
C. 1
D. cannot be determined

Answer

To solve the problem step by step, we need to analyze the given equations and find the values of \( x \) and \( y \).

### Step 1: Analyze the first equation
We have:
\[ \sin(4x - y) = 1 \]
The sine function equals 1 at \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). Therefore, we can set:
\[ 4x - y = 90^\circ \]

### Step 2: Analyze the second equation
Next, we have:
\[ \cos(2x + y) = \frac{1}{2} \]
The cosine function equals \( \frac{1}{2} \) at \( 60^\circ \) (or \( \frac{\pi}{3} \) radians). Thus, we can set:
\[ 2x + y = 60^\circ \]

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( 4x - y = 90^\circ \) (Equation 1)
2. \( 2x + y = 60^\circ \) (Equation 2)

We can solve these equations simultaneously.

First, we can express \( y \) from Equation 1:
\[ y = 4x - 90^\circ \]

Now, substitute \( y \) into Equation 2:
\[ 2x + (4x - 90^\circ) = 60^\circ \]
Combine like terms:
\[ 6x - 90^\circ = 60^\circ \]
Add \( 90^\circ \) to both sides:
\[ 6x = 150^\circ \]
Now, divide by 6:
\[ x = 25^\circ \]

### Step 4: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ y = 4(25^\circ) - 90^\circ = 100^\circ - 90^\circ = 10^\circ \]

### Step 5: Calculate \( \cot(x + 2y) \)
Now we need to find \( \cot(x + 2y) \):
\[ x + 2y = 25^\circ + 2(10^\circ) = 25^\circ + 20^\circ = 45^\circ \]
Thus,
\[ \cot(45^\circ) = 1 \]

### Final Answer
The value of \( \cot(x + 2y) \) is:
\[ \boxed{1} \]