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 compute \( u^2 + v^2 \) 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, let's expand \( u \):

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

Expanding the first part:

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

Using the product-to-sum identities, we know:

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

Thus, we can rewrite \( u \):

\[ u = 1 + \cos 2\theta + \cos \theta + \cos 3\theta \]

### Step 2: Simplify \( u \)

Now, we can combine like terms:

\[ u = 1 + \cos \theta + \cos 2\theta + \cos 3\theta \]

### Step 3: Expand \( v \)

Now let's expand \( v \):

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

Expanding 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 have:

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

Thus, we can rewrite \( v \):

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

Combining terms gives:

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

### Step 4: Compute \( u^2 + v^2 \)

Now we compute \( u^2 + v^2 \):

\[ u^2 + v^2 = (1 + \cos \theta + \cos 2\theta + \cos 3\theta)^2 + \left(\frac{3}{2}\sin \theta + \sin 2\theta + \frac{1}{2}\sin(3\theta)\right)^2 \]

Using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \), we can simplify this further.

### Step 5: Final Expression

After simplification, we find:

\[ u^2 + v^2 = 4 \left( \cos^2 \frac{3\theta}{2} + \sin^2 \frac{3\theta}{2} \right) = 4 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{4} \]