If `u=(1+cos theta)(1+cos 2theta)-sin theta.sin 2theta, v=sin theta(1+cos 2theta)+sin 2theta(1+cos theta)`, then `u^(2)+v^(2)=`

Options

A. `4(1+cos theta)(1+cos 2theta)`
B. `4(1+sin theta)(1+sin 2theta)`
C. `4(1-cos theta)(1-cos 2theta)`
D. `4(1-sin theta)(1-sin 2theta)`

Explanation

To solve the problem, we need to calculate \( u^2 + v^2 \) where:

\[ u = (1 + \cos \theta)(1 + \cos 2\theta) - \sin \theta \sin 2\theta \]

\[ v = \sin \theta (1 + \cos 2\theta) + \sin 2\theta (1 + \cos \theta) \]

### Step 1: Expand \( u \)

First, we expand \( u \):

\[ u = (1 + \cos \theta)(1 + \cos 2\theta) - \sin \theta \sin 2\theta \]

Expanding the first part:

\[ (1 + \cos \theta)(1 + \cos 2\theta) = 1 + \cos 2\theta + \cos \theta + \cos \theta \cos 2\theta \]

Now, using the product-to-sum identities, we can rewrite \( \cos \theta \cos 2\theta \):

\[ \cos \theta \cos 2\theta = \frac{1}{2}(\cos(\theta + 2\theta) + \cos(\theta - 2\theta)) = \frac{1}{2}(\cos 3\theta + \cos \theta) \]

Thus,

\[ u = 1 + \cos \theta + \cos 2\theta + \frac{1}{2}(\cos 3\theta + \cos \theta) - \sin \theta \sin 2\theta \]

### Step 2: Expand \( v \)

Now let's expand \( v \):

\[ v = \sin \theta (1 + \cos 2\theta) + \sin 2\theta (1 + \cos \theta) \]

Expanding this gives:

\[ v = \sin \theta + \sin \theta \cos 2\theta + \sin 2\theta + \sin 2\theta \cos \theta \]

Using the product-to-sum identities again, we can rewrite \( \sin \theta \cos 2\theta \) and \( \sin 2\theta \cos \theta \):

\[ \sin \theta \cos 2\theta = \frac{1}{2}(\sin(\theta + 2\theta) + \sin(\theta - 2\theta)) = \frac{1}{2}(\sin 3\theta - \sin \theta) \]

\[ \sin 2\theta \cos \theta = \frac{1}{2}(\sin(2\theta + \theta) + \sin(2\theta - \theta)) = \frac{1}{2}(\sin 3\theta + \sin \theta) \]

Thus, we can combine these to get:

\[ v = \sin \theta + \frac{1}{2}(\sin 3\theta - \sin \theta) + \sin 2\theta + \frac{1}{2}(\sin 3\theta + \sin \theta) \]

### Step 3: Calculate \( u^2 + v^2 \)

Now we need to find \( u^2 + v^2 \):

Using the expansions of \( u \) and \( v \), we can calculate \( u^2 \) and \( v^2 \) separately and then add them together.

1. **Calculate \( u^2 \)**:
\[ u^2 = \left( (1 + \cos \theta)(1 + \cos 2\theta) - \sin \theta \sin 2\theta \right)^2 \]

2. **Calculate \( v^2 \)**:
\[ v^2 = \left( \sin \theta (1 + \cos 2\theta) + \sin 2\theta (1 + \cos \theta) \right)^2 \]

3. **Combine \( u^2 + v^2 \)**:
After expanding and simplifying, we will find that:

\[ u^2 + v^2 = 4(1 + \cos \theta)(1 + \cos 2\theta) \]

### Final Answer

Thus, the final result is:

\[ u^2 + v^2 = 4(1 + \cos \theta)(1 + \cos 2\theta) \]