The value of the expression is

Options

A. 1
B. `-1`
C. 13
D. 0

Answer

To find the value of the expression \( 3(\sin \theta - \cos \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) \), we will simplify each part step by step.

### Step 1: Simplify \( 3(\sin \theta - \cos \theta)^4 \)

Using the binomial expansion:

\[ (\sin \theta - \cos \theta)^4 = \sin^4 \theta - 4\sin^3 \theta \cos \theta + 6\sin^2 \theta \cos^2 \theta - 4\sin \theta \cos^3 \theta + \cos^4 \theta \]

Thus,

\[ 3(\sin \theta - \cos \theta)^4 = 3(\sin^4 \theta + \cos^4 \theta - 4\sin^3 \theta \cos \theta + 6\sin^2 \theta \cos^2 \theta - 4\sin \theta \cos^3 \theta) \]

### Step 2: Simplify \( 6(\sin \theta + \cos \theta)^2 \)

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

\[ 6(\sin \theta + \cos \theta)^2 = 6(\sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta) = 6(1 + 2\sin \theta \cos \theta) = 6 + 12\sin \theta \cos \theta \]

### Step 3: Simplify \( 4(\sin^6 \theta + \cos^6 \theta) \)

Using the identity \( a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4) \):

\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) = 1(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]

Thus,

\[ 4(\sin^6 \theta + \cos^6 \theta) = 4(\sin^4 \theta + \cos^4 \theta - \sin^2 \theta \cos^2 \theta) \]

### Step 4: Combine all parts

Now we combine all the simplified parts:

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

Combine like terms:

- The coefficients of \( \sin^4 \theta + \cos^4 \theta \) become \( 3 + 4 = 7 \).
- The coefficients of \( \sin^2 \theta \cos^2 \theta \) become \( 18 - 4 = 14 \).
- The terms involving \( \sin^3 \theta \cos \theta \) and \( \sin \theta \cos^3 \theta \) cancel out.
- The constant term is \( 6 \).

### Final Expression

Thus, we have:

\[ 7(\sin^4 \theta + \cos^4 \theta) + 14\sin^2 \theta \cos^2 \theta + 6 \]

Using the identity \( \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \):

\[ = 7(1 - 2\sin^2 \theta \cos^2 \theta) + 14\sin^2 \theta \cos^2 \theta + 6 \]

This simplifies to:

\[ 7 - 14\sin^2 \theta \cos^2 \theta + 14\sin^2 \theta \cos^2 \theta + 6 = 7 + 6 = 13 \]

### Conclusion

Thus, the value of the expression is:

\[ \boxed{13} \]