The value of `8(sin^(6)theta+cos^(6)theta)-12(sin^(4)theta+cos^(4)theta)`is equal to

Options

A. 20
B. `-20`
C. `-4`
D. 4

Answer

To solve the expression \( 8(\sin^6 \theta + \cos^6 \theta) - 12(\sin^4 \theta + \cos^4 \theta) \), we can use some algebraic identities and simplifications.

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

We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \).

Thus, we have:
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)((\sin^2 \theta)^2 - \sin^2 \theta \cos^2 \theta + (\cos^2 \theta)^2) \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), this simplifies to:
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta)^2 - \sin^2 \theta \cos^2 \theta + (\cos^2 \theta)^2 \]
Now, we can express \( \sin^4 \theta + \cos^4 \theta \) using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 2: Substitute back into the expression

Now substituting these identities into the original expression:
\[ 8(\sin^6 \theta + \cos^6 \theta) - 12(\sin^4 \theta + \cos^4 \theta) \]
becomes:
\[ 8\left((\sin^2 \theta)^2 - \sin^2 \theta \cos^2 \theta + (\cos^2 \theta)^2\right) - 12(1 - 2\sin^2 \theta \cos^2 \theta) \]

### Step 3: Simplify the expression

Now we can simplify:
1. The term \( (\sin^2 \theta)^2 + (\cos^2 \theta)^2 \) can be rewritten as \( 1 - 2\sin^2 \theta \cos^2 \theta \).
2. Thus, we have:
\[ 8\left(1 - 3\sin^2 \theta \cos^2 \theta\right) - 12 + 24\sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ 8 - 12 + 8\sin^2 \theta \cos^2 \theta = -4 + 16\sin^2 \theta \cos^2 \theta \]

### Step 4: Use the double angle identity

Using the double angle identity \( \sin^2 \theta \cos^2 \theta = \frac{1}{4}\sin^2(2\theta) \):
\[ 16\sin^2 \theta \cos^2 \theta = 4\sin^2(2\theta) \]
Thus, the expression becomes:
\[ -4 + 4\sin^2(2\theta) \]

### Final Answer

Therefore, the final value of the expression is:
\[ 4\sin^2(2\theta) - 4 \]