For any `theta epsilon((pi)/4,(pi)/2)` the expression `3(sin theta-cos theta)^(4)+6(sin theta+cos theta)^(2)+4sin^(6)theta` equals:

Options

A. `13-4cos^(2)+6sin^(2) theta cos^(2)theta`
B. `13-4cos^(4) theta+2sin^(2) theta cos^(2) theta`
C. `13-4cos^(6)theta`
D. `13-4cos^(2) theta+6cos^(4) theta`

Explanation

To solve the expression \(3(\sin \theta - \cos \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4\sin^6 \theta\) for \(\theta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)\), we will break it down step by step.

### Step 1: Simplify \((\sin \theta - \cos \theta)^2\) and \((\sin \theta + \cos \theta)^2\)

Using the identity \((a - b)^2 = a^2 - 2ab + b^2\) and \((a + b)^2 = a^2 + 2ab + b^2\):

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

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

### Step 2: Substitute into the expression

Now we substitute these results back into the original expression:

\[ 3(\sin \theta - \cos \theta)^4 = 3(1 - \sin(2\theta))^2 \]

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

### Step 3: Expand \((1 - \sin(2\theta))^2\)

Expanding \(3(1 - \sin(2\theta))^2\):

\[ 3(1 - \sin(2\theta))^2 = 3(1 - 2\sin(2\theta) + \sin^2(2\theta)) = 3 - 6\sin(2\theta) + 3\sin^2(2\theta) \]

### Step 4: Combine the terms

Now combine all parts of the expression:

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

Notice that \(-6\sin(2\theta)\) and \(6\sin(2\theta)\) cancel each other out:

\[ = 9 + 3\sin^2(2\theta) + 4\sin^6(\theta) \]

### Step 5: Use the identity for \(\sin^6(\theta)\)

We can express \(\sin^6(\theta)\) in terms of \(\sin^2(\theta)\):

\[ \sin^6(\theta) = (\sin^2(\theta))^3 \]

Let \(x = \sin^2(\theta)\). Then, \(\sin^2(2\theta) = 4\sin^2(\theta)\cos^2(\theta) = 4x(1-x)\).

### Step 6: Substitute back

Substituting back gives:

\[ = 9 + 3(4x(1-x)) + 4x^3 \]

\[ = 9 + 12x - 12x^2 + 4x^3 \]

### Step 7: Final expression

Thus, the final expression simplifies to:

\[ = 4x^3 - 12x^2 + 12x + 9 \]

### Conclusion

The expression \(3(\sin \theta - \cos \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4\sin^6 \theta\) simplifies to \(4\sin^6(\theta) - 12\sin^4(\theta) + 12\sin^2(\theta) + 9\).