Given that (sin(x) + cos(x))^2 = p, express the following in terms of p:
1. sin(x) - cos(x)
2. sin^3(x) + cos^3(x)

Given the equation \((\sin(x) + \cos(x))^2 = p\), we need to express the following in terms of \(p\):

1. **Expression for \(\sin(x) - \cos(x)\):**

   We start by expanding \((\sin(x) + \cos(x))^2\):
   \[
   (\sin(x) + \cos(x))^2 = \sin^2(x) + 2\sin(x)\cos(x) + \cos^2(x)
   \]
   Since \(\sin^2(x) + \cos^2(x) = 1\), we can substitute this into the equation:
   \[
   p = 1 + 2\sin(x)\cos(x)
   \]
   We also know the identity:
   \[
   \sin(2x) = 2\sin(x)\cos(x)
   \]
   Therefore, we can rewrite \(p\) as:
   \[
   p = 1 + \sin(2x)
   \]
   Now, let's find an expression for \(\sin(x) - \cos(x)\). We start by expanding \((\sin(x) - \cos(x))^2\):
   \[
   (\sin(x) - \cos(x))^2 = \sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x)
   \]
   Again, using \(\sin^2(x) + \cos^2(x) = 1\), we have:
   \[
   (\sin(x) - \cos(x))^2 = 1 - 2\sin(x)\cos(x)
   \]
   Substituting \(\sin(2x) = 2\sin(x)\cos(x)\), this becomes:
   \[
   (\sin(x) - \cos(x))^2 = 1 - \sin(2x)
   \]
   Since \(p = 1 + \sin(2x)\), we can express \(\sin(2x)\) as:
   \[
   \sin(2x) = p - 1
   \]
   Substituting this into the equation for \((\sin(x) - \cos(x))^2\):
   \[
   (\sin(x) - \cos(x))^2 = 1 - (p - 1) = 2 - p
   \]
   Therefore:
   \[
   \sin(x) - \cos(x) = \pm\sqrt{2 - p}
   \]

2. **Expression for \(\sin^3(x) + \cos^3(x)\):**

   We can use the identity:
   \[
   \sin^3(x) + \cos^3(x) = (\sin(x) + \cos(x))(\sin^2(x) - \sin(x)\cos(x) + \cos^2(x))
   \]
   First, we simplify the expression \(\sin^2(x) - \sin(x)\cos(x) + \cos^2(x)\):
   \[
   \sin^2(x) + \cos^2(x) = 1
   \]
   \[
   \sin^2(x) - \sin(x)\cos(x) + \cos^2(x) = 1 - \sin(x)\cos(x)
   \]
   Since \(\sin(x)\cos(x) = \frac{p - 1}{2}\), this becomes:
   \[
   \sin^2(x) - \sin(x)\cos(x) + \cos^2(x) = 1 - \frac{p - 1}{2} = \frac{3 - p}{2}
   \]
   Now, multiply by \(\sin(x) + \cos(x)\):
   \[
   \sin^3(x) + \cos^3(x) = (\sin(x) + \cos(x)) \times \frac{3 - p}{2}
   \]
   Since \(\sin(x) + \cos(x) = \sqrt{p}\), the expression becomes:
   \[
   \sin^3(x) + \cos^3(x) = \sqrt{p} \times \frac{3 - p}{2}
   \]
   Therefore:
   \[
   \sin^3(x) + \cos^3(x) = \frac{\sqrt{p}(3 - p)}{2}
   \]

Would you like more details or have any questions?

Here are some related questions:
1. How can we derive \(\sin(x)\cos(x)\) using trigonometric identities?
2. What is the geometric interpretation of the identity \((\sin(x) + \cos(x))^2 = p\)?
3. Can \(\sin^3(x) + \cos^3(x)\) be further simplified under special conditions?
4. How does the range of \(p\) affect the values of \(\sin(x) - \cos(x)\)?
5. What happens when \(p = 2\) in the given expressions?

**Tip:** Trigonometric identities are powerful tools for transforming and simplifying expressions. Always look for opportunities to use them in problems involving trigonometric functions.

Explore how to transform trigonometric expressions using the identity (sin(x) + cos(x))^2 = p. Learn how to express sin(x) - cos(x) and sin^3(x) + cos^3(x) in terms of p with detailed step-by-step solutions. Suitable for high school students studying trigonometry.

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