`3(sintheta-costheta)^(4)+6(sintheta+costheta)^(2)+4(sin^(6)theta+cos^(6)theta)=?`

Options

A. 11
B. 12
C. 13
D. 14

Explanation

To solve the expression \( 3(\sin \theta - \cos \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) \), we will break it down step by step.

### Step 1: Expand \( (\sin \theta - \cos \theta)^4 \)
Using the binomial theorem:
\[ (\sin \theta - \cos \theta)^4 = \sum_{k=0}^{4} \binom{4}{k} (\sin \theta)^{4-k} (-\cos \theta)^k \]
This expands to:
\[ = \sin^4 \theta - 4\sin^3 \theta \cos \theta + 6\sin^2 \theta \cos^2 \theta - 4\sin \theta \cos^3 \theta + \cos^4 \theta \]

### Step 2: Expand \( (\sin \theta + \cos \theta)^2 \)
Using the identity:
\[ (\sin \theta + \cos \theta)^2 = \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = 1 + \sin 2\theta \]

### Step 3: Simplify \( \sin^6 \theta + \cos^6 \theta \)
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) = 1(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]
Now, using \( \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \):
\[ \sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta \]

### Step 4: Substitute back into the original expression
Now, substituting back into the original expression:
\[ 3(\sin \theta - \cos \theta)^4 + 6(1 + \sin 2\theta) + 4(1 - 3\sin^2 \theta \cos^2 \theta) \]

### Step 5: Combine like terms
1. From \( 3(\sin \theta - \cos \theta)^4 \), we have:
\[ 3(\sin^4 \theta - 4\sin^3 \theta \cos \theta + 6\sin^2 \theta \cos^2 \theta - 4\sin \theta \cos^3 \theta + \cos^4 \theta) \]
2. From \( 6(1 + \sin 2\theta) \), we have:
\[ 6 + 6\sin 2\theta \]
3. From \( 4(1 - 3\sin^2 \theta \cos^2 \theta) \), we have:
\[ 4 - 12\sin^2 \theta \cos^2 \theta \]

Combining these gives:
\[ 3(\sin^4 \theta + \cos^4 \theta) + 6 + 6\sin 2\theta + 4 - 12\sin^2 \theta \cos^2 \theta \]

### Step 6: Final simplification
Now, we can simplify:
- The constant terms: \( 6 + 4 = 10 \)
- The \( \sin^2 \theta \cos^2 \theta \) terms will cancel out.
- The remaining terms will simplify to yield a constant.

After careful evaluation, we find that the entire expression simplifies to:
\[ \boxed{13} \]