The expression `3[sin^4(3/2pi-alpha)+sin^4(3pi+alpha)]-2[sin^6(1/2pi+alpha)+sin^6(5pi-alpha)]` is equal to

Options

A. 0
B. 1
C. 3
D. None of these

Explanation

To solve the expression \( 3[\sin^4(3/2\pi - \alpha) + \sin^4(3\pi + \alpha)] - 2[\sin^6(1/2\pi + \alpha) + \sin^6(5\pi - \alpha)] \), we will simplify each term step by step.

### Step 1: Simplify \(\sin(3/2\pi - \alpha)\) and \(\sin(3\pi + \alpha)\)

Using the sine subtraction and addition formulas:
- \(\sin(3/2\pi - \alpha) = \cos(\alpha)\)
- \(\sin(3\pi + \alpha) = -\sin(\alpha)\)

Thus, we have:
\[ \sin^4(3/2\pi - \alpha) = \cos^4(\alpha) \]
\[ \sin^4(3\pi + \alpha) = \sin^4(\alpha) \]

### Step 2: Substitute into the expression

Now substituting these values into the expression:
\[ 3[\cos^4(\alpha) + \sin^4(\alpha)] \]

### Step 3: Simplify \(\sin(1/2\pi + \alpha)\) and \(\sin(5\pi - \alpha)\)

Using the sine addition and subtraction formulas:
- \(\sin(1/2\pi + \alpha) = \cos(\alpha)\)
- \(\sin(5\pi - \alpha) = -\sin(\alpha)\)

Thus, we have:
\[ \sin^6(1/2\pi + \alpha) = \cos^6(\alpha) \]
\[ \sin^6(5\pi - \alpha) = \sin^6(\alpha) \]

### Step 4: Substitute into the expression

Now substituting these values into the expression:
\[ -2[\cos^6(\alpha) + \sin^6(\alpha)] \]

### Step 5: Combine the terms

Now we can rewrite the entire expression:
\[ 3[\cos^4(\alpha) + \sin^4(\alpha)] - 2[\cos^6(\alpha) + \sin^6(\alpha)] \]

### Step 6: Use the identity for \(\cos^4(\alpha) + \sin^4(\alpha)\)

We know that:
\[ \cos^4(\alpha) + \sin^4(\alpha) = (\cos^2(\alpha) + \sin^2(\alpha))^2 - 2\cos^2(\alpha)\sin^2(\alpha) = 1 - 2\cos^2(\alpha)\sin^2(\alpha) \]

### Step 7: Use the identity for \(\cos^6(\alpha) + \sin^6(\alpha)\)

We can also use:
\[ \cos^6(\alpha) + \sin^6(\alpha) = (\cos^2(\alpha) + \sin^2(\alpha))(\cos^4(\alpha) - \cos^2(\alpha)\sin^2(\alpha) + \sin^4(\alpha)) = 1(1 - 3\cos^2(\alpha)\sin^2(\alpha)) \]

### Step 8: Substitute back into the expression

Substituting these identities back into our expression gives:
\[ 3[1 - 2\cos^2(\alpha)\sin^2(\alpha)] - 2[1 - 3\cos^2(\alpha)\sin^2(\alpha)] \]

### Step 9: Simplify the expression

Now simplifying:
\[ = 3 - 6\cos^2(\alpha)\sin^2(\alpha) - 2 + 6\cos^2(\alpha)\sin^2(\alpha) \]
\[ = 1 \]

### Final Result

Thus, the value of the expression is:
\[ \boxed{1} \]