`3 (sinx - cosx)^(4) + 6 (sinx + cosx)^(2) + 4(sin^(6)x + cos^(6)x) = `

Options

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

Answer

To solve the equation \(3 (\sin x - \cos x)^4 + 6 (\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x)\), we will break it down into three parts and simplify each one step by step.

### Step 1: Simplifying \(3 (\sin x - \cos x)^4\)

1. **Expand the expression**:
\[ (\sin x - \cos x)^4 = (\sin^2 x - 2\sin x \cos x + \cos^2 x)^2 \]
Since \(\sin^2 x + \cos^2 x = 1\):
\[ = (1 - 2\sin x \cos x)^2 \]
Now, expand this:
\[ = 1 - 4\sin x \cos x + 4\sin^2 x \cos^2 x \]

2. **Multiply by 3**:
\[ 3(\sin x - \cos x)^4 = 3(1 - 4\sin x \cos x + 4\sin^2 x \cos^2 x) = 3 - 12\sin x \cos x + 12\sin^2 x \cos^2 x \]

### Step 2: Simplifying \(6 (\sin x + \cos x)^2\)

1. **Expand the expression**:
\[ (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x \]
Again, using \(\sin^2 x + \cos^2 x = 1\):
\[ = 1 + 2\sin x \cos x \]

2. **Multiply by 6**:
\[ 6(\sin x + \cos x)^2 = 6(1 + 2\sin x \cos x) = 6 + 12\sin x \cos x \]

### Step 3: Simplifying \(4(\sin^6 x + \cos^6 x)\)

1. **Use the identity for sum of cubes**:
\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]
Since \(\sin^2 x + \cos^2 x = 1\):
\[ = \sin^4 x - \sin^2 x \cos^2 x + \cos^4 x \]

2. **Use the identity for squares**:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]
Therefore:
\[ \sin^6 x + \cos^6 x = 1 - 3\sin^2 x \cos^2 x \]

3. **Multiply by 4**:
\[ 4(\sin^6 x + \cos^6 x) = 4(1 - 3\sin^2 x \cos^2 x) = 4 - 12\sin^2 x \cos^2 x \]

### Step 4: Combine all parts

Now, we combine all three parts:
\[ 3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x) \]
Substituting the simplified expressions:
\[ = (3 - 12\sin x \cos x + 12\sin^2 x \cos^2 x) + (6 + 12\sin x \cos x) + (4 - 12\sin^2 x \cos^2 x) \]

### Step 5: Simplify the final expression

Combine like terms:
\[ = 3 + 6 + 4 + (-12\sin x \cos x + 12\sin x \cos x) + (12\sin^2 x \cos^2 x - 12\sin^2 x \cos^2 x) \]
This simplifies to:
\[ = 13 \]

### Final Answer
Thus, the final result is:
\[ \boxed{13} \]