`4(sin^(6)theta+cos^(6)theta)-6(sin^(4)theta+cos^(4)theta)`is equal to

Options

A. `0`
B. `1`
C. `-2`
D. none of these

Explanation

To solve the expression \( 4(\sin^6 \theta + \cos^6 \theta) - 6(\sin^4 \theta + \cos^4 \theta) \), we can use some algebraic identities and simplifications. Let's break it down step by step.

### Step 1: Use the identity for \( \sin^6 \theta + \cos^6 \theta \)

We know that:
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \sin^6 \theta + \cos^6 \theta = \sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta \]

### Step 2: Substitute into the expression

Now, substituting this into our original expression:
\[ 4(\sin^6 \theta + \cos^6 \theta) = 4(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]

### Step 3: Combine terms

Now, we can rewrite the expression:
\[ 4(\sin^4 \theta + \cos^4 \theta) - 4\sin^2 \theta \cos^2 \theta - 6(\sin^4 \theta + \cos^4 \theta) \]
This simplifies to:
\[ (4 - 6)(\sin^4 \theta + \cos^4 \theta) - 4\sin^2 \theta \cos^2 \theta = -2(\sin^4 \theta + \cos^4 \theta) - 4\sin^2 \theta \cos^2 \theta \]

### Step 4: Use the identity for \( \sin^4 \theta + \cos^4 \theta \)

We can use the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 5: Substitute back

Substituting this back into our expression gives:
\[ -2(1 - 2\sin^2 \theta \cos^2 \theta) - 4\sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ -2 + 4\sin^2 \theta \cos^2 \theta - 4\sin^2 \theta \cos^2 \theta = -2 \]

### Final Result

Thus, the final result of the expression \( 4(\sin^6 \theta + \cos^6 \theta) - 6(\sin^4 \theta + \cos^4 \theta) \) is:
\[ \boxed{-2} \]