If `x+y=3-cos 4 theta` and `x-y=4sin 2 theta`,the value of `sqrtx+sqrty` is equal to

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we start with the given equations:

1. \( x + y = 3 - \cos(4\theta) \) (Equation 1)
2. \( x - y = 4 \sin(2\theta) \) (Equation 2)

### Step 1: Add the two equations
We can add Equation 1 and Equation 2 to eliminate \( y \):

\[ (x + y) + (x - y) = (3 - \cos(4\theta)) + (4 \sin(2\theta) \]

This simplifies to:

\[ 2x = 3 - \cos(4\theta) + 4 \sin(2\theta) \]

### Step 2: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{3 - \cos(4\theta) + 4 \sin(2\theta)}{2} \]

### Step 3: Subtract the two equations
Next, we subtract Equation 2 from Equation 1 to eliminate \( x \):

\[ (x + y) - (x - y) = (3 - \cos(4\theta)) - (4 \sin(2\theta)) \]

This simplifies to:

\[ 2y = 3 - \cos(4\theta) - 4 \sin(2\theta) \]

### Step 4: Solve for \( y \)
Now, we can solve for \( y \):

\[ y = \frac{3 - \cos(4\theta) - 4 \sin(2\theta)}{2} \]

### Step 5: Find \( \sqrt{x} + \sqrt{y} \)
Now we need to find \( \sqrt{x} + \sqrt{y} \). We first express \( \sqrt{x} \) and \( \sqrt{y} \):

\[ \sqrt{x} = \sqrt{\frac{3 - \cos(4\theta) + 4 \sin(2\theta)}{2}} \]
\[ \sqrt{y} = \sqrt{\frac{3 - \cos(4\theta) - 4 \sin(2\theta)}{2}} \]

Now we can add these two expressions:

\[ \sqrt{x} + \sqrt{y} = \sqrt{\frac{3 - \cos(4\theta) + 4 \sin(2\theta)}{2}} + \sqrt{\frac{3 - \cos(4\theta) - 4 \sin(2\theta)}{2}} \]

### Step 6: Simplify the expression
To simplify, we can factor out \( \sqrt{\frac{1}{2}} \):

\[ \sqrt{x} + \sqrt{y} = \sqrt{\frac{1}{2}} \left( \sqrt{3 - \cos(4\theta) + 4 \sin(2\theta)} + \sqrt{3 - \cos(4\theta) - 4 \sin(2\theta)} \right) \]

### Step 7: Use the identity for simplification
Using the identity \( a^2 + b^2 + 2ab = (a + b)^2 \) and \( a^2 + b^2 - 2ab = (a - b)^2 \), we can see that:

\[ \sqrt{3 - \cos(4\theta) + 4 \sin(2\theta)} + \sqrt{3 - \cos(4\theta) - 4 \sin(2\theta)} = 2 \]

Thus, we have:

\[ \sqrt{x} + \sqrt{y} = \sqrt{\frac{1}{2}} \cdot 2 = \sqrt{2} \]

### Conclusion
The value of \( \sqrt{x} + \sqrt{y} \) is \( 2 \).