The value of 4(sin^6 x + cos^6 x) + 3 sin^2 2x is

To find the value of the expression:

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

let's simplify it step by step.

### Step 1: Use algebraic identities

First, notice that we can use the identity:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

For \( a = \sin^2 x \) and \( b = \cos^2 x \), we have:

\[
\sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x) \left( (\sin^2 x)^2 - \sin^2 x \cos^2 x + (\cos^2 x)^2 \right)
\]

Since \( \sin^2 x + \cos^2 x = 1 \), we get:

\[
\sin^6 x + \cos^6 x = 1 \cdot \left( \sin^4 x - \sin^2 x \cos^2 x + \cos^4 x \right)
\]

Now, observe that:

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

Thus,

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

### Step 2: Use trigonometric identities

Next, consider the term \( \sin^2 2x \). Recall that:

\[
\sin 2x = 2 \sin x \cos x \implies \sin^2 2x = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x
\]

So, \( \sin^2 2x = 4 \sin^2 x \cos^2 x \).

### Step 3: Substitute back into the expression

Now substitute these identities back into the original expression:

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

Simplify further:

\[
= 4 - 12 \sin^2 x \cos^2 x + 12 \sin^2 x \cos^2 x
\]

The terms \( -12 \sin^2 x \cos^2 x \) and \( +12 \sin^2 x \cos^2 x \) cancel each other out, leaving:

\[
= 4
\]

### Conclusion

The value of the expression is:

\[
\boxed{4}
\]

Would you like further details or have any questions? 

### Related Questions

1. How do we derive the formula for \( \sin^2 2x \) using trigonometric identities?
2. What other algebraic identities are useful in simplifying trigonometric expressions?
3. How can we use the identity \( \sin^2 x + \cos^2 x = 1 \) in different types of problems?
4. What are some common applications of trigonometric identities in calculus?
5. How can we derive the triple angle formulas using basic trigonometric identities?

**Tip:** When dealing with trigonometric expressions, always look for identities or factorizations that can simplify your work!

Discover how to evaluate the expression 4(sin^6 x + cos^6 x) + 3 sin^2 2x with a detailed step-by-step solution. This problem involves trigonometric identities and algebraic simplification techniques, suitable for high school students.

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