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

Options

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

Answer

To solve the problem, we have the following equations:

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

We need to find \( \cos^2(\theta - \phi) \).

### Step 1: Square the first equation

Starting with the first equation:

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

Squaring both sides gives:

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

Expanding the left-hand side:

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

### Step 2: Square the second equation

Now, squaring the second equation:

\[ \cos 2\theta + \cos 2\phi = \frac{3}{2} \]

Squaring both sides gives:

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

Expanding the left-hand side:

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

### Step 3: Use the Pythagorean identity

Using the identity \( \sin^2 x + \cos^2 x = 1 \):

From the first equation, we have:

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

From the second equation, we have:

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

Adding these two results together:

\[ (\sin^2 2\theta + \cos^2 2\theta) + (\sin^2 2\phi + \cos^2 2\phi) = \left(\frac{1}{4} - 2 \sin 2\theta \sin 2\phi\right) + \left(\frac{9}{4} - 2 \cos 2\theta \cos 2\phi\right) \]

This simplifies to:

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

### Step 4: Simplify and solve for \( \cos^2(\theta - \phi) \)

Using the cosine of the difference identity:

\[ \cos(2\theta - 2\phi) = \cos 2\theta \cos 2\phi + \sin 2\theta \sin 2\phi \]

Let \( x = \cos(2\theta - 2\phi) \):

We can rewrite our equation as:

\[ 2 = \frac{10}{4} - 2x \]

Solving for \( x \):

\[ 2x = \frac{10}{4} - 2 \]

\[ 2x = \frac{10}{4} - \frac{8}{4} = \frac{2}{4} = \frac{1}{2} \]

\[ x = \frac{1}{4} \]

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

Now, using the identity:

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

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

\[ \cos^2(\theta - \phi) = \frac{1 + \frac{1}{4}}{2} = \frac{\frac{5}{4}}{2} = \frac{5}{8} \]

Thus, the final answer is:

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