If sin^4 A/a + cos^4 A/b = 1/a+b, then sin^8 A/a^3 + cos^8 A/b^3 is?

We are given the equation:

\[
\frac{\sin^4 A}{a} + \frac{\cos^4 A}{b} = \frac{1}{a + b}
\]

We need to find the value of:

\[
\frac{\sin^8 A}{a^3} + \frac{\cos^8 A}{b^3}
\]

### Step 1: Simplify the given equation
Start by multiplying both sides of the given equation by \( a + b \):

\[
(a + b) \left( \frac{\sin^4 A}{a} + \frac{\cos^4 A}{b} \right) = 1
\]

This becomes:

\[
(a + b) \left( \frac{\sin^4 A}{a} + \frac{\cos^4 A}{b} \right) = 1
\]

Multiply out each term:

\[
\sin^4 A \cdot \frac{a + b}{a} + \cos^4 A \cdot \frac{a + b}{b} = 1
\]

Simplifying each term:

\[
\sin^4 A \left( 1 + \frac{b}{a} \right) + \cos^4 A \left( 1 + \frac{a}{b} \right) = 1
\]

We now have a simplified expression. However, based on the original structure, we now seek to find \(\frac{\sin^8 A}{a^3} + \frac{\cos^8 A}{b^3}\).

#### Applying symmetry in \(a\) and \(b\)

If sin^4 A/a + cos^4 A/b = 1/(a+b), then sin^8 A/a^3 + cos^8 A/b^3 is?

This problem involves using a trigonometric identity and algebraic manipulation to find the value of sin^8 A/a^3 + cos^8 A/b^3, given that sin^4 A/a + cos^4 A/b equals 1/(a+b). A step-by-step approach simplifies the problem for students in Grades 11-12.

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