https://file.aiquickdraw.com/mathbot/file/1726148105510pdy4f0hz.jpg

To solve the expression provided in the image, let's define it clearly:

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

We'll approach this by expanding and simplifying each term step-by-step.

### Step 1: Simplify the terms

1. **Term 1: \(6(\sin \theta + \cos \theta)^2\)**

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

Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

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

So,

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

2. **Term 2: \(3(\sin \theta - \cos \theta)^4\)**

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

So,

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

Expanding the square:

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

Thus,

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

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

3. **Term 3: \(4(\sin^6 \theta + \cos^6 \theta)\)**

Using the identity:

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

Since \(\sin^2 \theta + \cos^2 \theta = 1\):

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

Thus,

\[
4(\sin^6 \theta + \cos^6 \theta) = 4(1 - 3 \sin^2 \theta \cos^2 \theta) = 4 - 12 (\sin \theta \cos \theta)^2
\]

### Step 2: Combine all terms

Now we can add up all three terms:

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

Combine like terms:

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

This simplifies to:

\[
13 + 0 + 0 = 13
\]

### Final Answer

The value of the expression is **13**.

Would you like any further details or have any questions? 

Here are some related questions that might interest you:

1. What are other trigonometric identities useful for simplifying expressions?
2. How can we use complex numbers to represent trigonometric functions?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How do we derive the double-angle and half-angle formulas?
5. How do we solve trigonometric equations involving multiple angles?

**Tip:** Always look for identities and patterns when dealing with trigonometric expressions to simplify them effectively.

The value of 6(sin θ + cos θ)² + 3(sin θ − cos θ)⁴ + 4(sin⁶ θ + cos⁶ θ) is

Learn how to simplify the trigonometric expression 6(sin θ + cos θ)² + 3(sin θ − cos θ)⁴ + 4(sin⁶ θ + cos⁶ θ) with a detailed step-by-step solution. This problem covers key trigonometric concepts, including identities and simplification techniques. Suitable for Grades 11-12.

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