We need to prove the following expression:

$\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} + \frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} = 2$

Step 1: Simplify each fraction using algebraic identities

We use the algebraic identities for the sum and difference of cubes:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Substitute $a = \sin \theta$ and $b = \cos \theta$:

$\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)$ $\sin^3 \theta - \cos^3 \theta = (\sin \theta - \cos \theta)(\sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta)$

Step 2: Substitute into the original expression

Now, substitute these identities into the original expression:

$\frac{(\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)}{\sin \theta + \cos \theta} + \frac{(\sin \theta - \cos \theta)(\sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta)}{\sin \theta - \cos \theta}$

The $\sin \theta + \cos \theta$ and $\sin \theta - \cos \theta$ terms in the numerators and denominators cancel out:

$\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta + \sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta$

Step 3: Simplify the resulting expression

Combine like terms:

$(\sin^2 \theta + \cos^2 \theta) + (\sin^2 \theta + \cos^2 \theta)$

We know that:

$\sin^2 \theta + \cos^2 \theta = 1$

So, the expression simplifies to:

$1 + 1 = 2$

Thus, the original expression equals $2$.

Final Result

$\boxed{2}$

The proof is complete.

---

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the properties of cubic functions?
2. How do the sum and difference of cubes identities work?
3. Can you prove other trigonometric identities using algebraic methods?
4. What are some common techniques for simplifying trigonometric expressions?
5. How do trigonometric identities help in solving equations?

Tip: Always look for algebraic identities that can simplify complex trigonometric expressions.