If `sin 2theta+sin 2phi=1//2` and `cos2theta+cos 2phi=3//2`, then `cos^(2)(theta-phi)=`

Options

A. `3//8`
B. `5//8`
C. `3//4`
D. `5//4`

Explanation

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

1. \( \sin 2\theta + \sin 2\phi = \frac{1}{2} \)
2. \( \cos 2\theta + \cos 2\phi = \frac{3}{2} \)

### Step 1: Square both equations

First, we square both equations to make use of the identities.

\[ (\sin 2\theta + \sin 2\phi)^2 = \left(\frac{1}{2}\right)^2 \]

This gives us:

\[ \sin^2 2\theta + \sin^2 2\phi + 2 \sin 2\theta \sin 2\phi = \frac{1}{4} \]

Similarly, for the second equation:

\[ (\cos 2\theta + \cos 2\phi)^2 = \left(\frac{3}{2}\right)^2 \]

This gives us:

\[ \cos^2 2\theta + \cos^2 2\phi + 2 \cos 2\theta \cos 2\phi = \frac{9}{4} \]

### Step 2: Write down the equations

Now we have two equations:

1. \( \sin^2 2\theta + \sin^2 2\phi + 2 \sin 2\theta \sin 2\phi = \frac{1}{4} \) (Equation 1)
2. \( \cos^2 2\theta + \cos^2 2\phi + 2 \cos 2\theta \cos 2\phi = \frac{9}{4} \) (Equation 2)

### Step 3: Use the Pythagorean Identity

Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can rewrite the equations:

From Equation 1:

\[ \sin^2 2\theta + \cos^2 2\theta + \sin^2 2\phi + \cos^2 2\phi + 2 \sin 2\theta \sin 2\phi + 2 \cos 2\theta \cos 2\phi = \frac{1}{4} + \frac{9}{4} \]

This simplifies to:

\[ 1 + 1 + 2(\sin 2\theta \sin 2\phi + \cos 2\theta \cos 2\phi) = \frac{10}{4} \]

### Step 4: Simplify further

Now, we can simplify:

\[ 2 + 2 \cos(2\theta - 2\phi) = \frac{10}{4} \]

This implies:

\[ 2 \cos(2\theta - 2\phi) = \frac{10}{4} - 2 \]

\[ 2 \cos(2\theta - 2\phi) = \frac{10 - 8}{4} = \frac{2}{4} = \frac{1}{2} \]

Dividing both sides by 2:

\[ \cos(2\theta - 2\phi) = \frac{1}{4} \]

### Step 5: Find \( \cos^2(\theta - \phi) \)

Using the double angle formula, we know:

\[ \cos(2x) = 2\cos^2(x) - 1 \]

Setting \( x = \theta - \phi \):

\[ \cos(2(\theta - \phi)) = 2\cos^2(\theta - \phi) - 1 \]

Substituting \( \cos(2(\theta - \phi)) = \frac{1}{4} \):

\[ \frac{1}{4} = 2\cos^2(\theta - \phi) - 1 \]

### Step 6: Solve for \( \cos^2(\theta - \phi) \)

Rearranging gives:

\[ 2\cos^2(\theta - \phi) = \frac{1}{4} + 1 \]

\[ 2\cos^2(\theta - \phi) = \frac{1}{4} + \frac{4}{4} = \frac{5}{4} \]

Dividing by 2:

\[ \cos^2(\theta - \phi) = \frac{5}{8} \]

### Final Answer

Thus, the value of \( \cos^2(\theta - \phi) \) is:

\[ \boxed{\frac{5}{8}} \]