Let us consider the equation `cos^4x/a+sin^4x/b=1/(a+b),x in[0,pi/2],a,bgt0`
the value of `sin^8x/b^3+cos^8x/a^3` is

Options

A. `sin^4x/b=cos^4x/a`
B. `sinx/a=cosx/b`
C. `sin^4x/b^2=cos^4x/a^2`
D. `sin^2x/a=cos^2x/b`

Explanation

To solve the equation

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

for \( x \in [0, \frac{\pi}{2}] \) and \( a, b > 0 \), we want to find the value of

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

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \frac{\cos^4 x}{a} + \frac{\sin^4 x}{b} = \frac{1}{a+b}. \]

2. **Multiply through by \( a + b \)**:
\[ (a + b) \left( \frac{\cos^4 x}{a} + \frac{\sin^4 x}{b} \right) = 1. \]
This simplifies to:
\[ \frac{(a+b) \cos^4 x}{a} + \frac{(a+b) \sin^4 x}{b} = 1. \]

3. **Distributing the terms**:
\[ \frac{b \cos^4 x + (a+b) \sin^4 x}{b} = 1. \]
Rearranging gives:
\[ b \cos^4 x + (a+b) \sin^4 x = b. \]

4. **Rearranging further**:
\[ b \cos^4 x = b - (a+b) \sin^4 x. \]

5. **Dividing through by \( b \)**:
\[ \cos^4 x = 1 - \frac{(a+b) \sin^4 x}{b}. \]

6. **Let \( \sin^2 x = y \)**, then \( \cos^2 x = 1 - y \):
\[ (1 - y)^2 = 1 - \frac{(a+b)y^2}{b}. \]

7. **Expanding and simplifying**:
\[ 1 - 2y + y^2 = 1 - \frac{(a+b)y^2}{b}. \]
Rearranging gives:
\[ -2y + y^2 + \frac{(a+b)y^2}{b} = 0. \]

8. **Combining like terms**:
\[ y^2 \left(1 + \frac{(a+b)}{b}\right) - 2y = 0. \]
Factoring out \( y \):
\[ y \left( y + \frac{(a+b)}{b} - 2 \right) = 0. \]

9. **Finding \( y \)**:
The solutions are \( y = 0 \) or \( y = 2 - \frac{(a+b)}{b} \).

10. **Substituting back to find \( \sin^8 x \) and \( \cos^8 x \)**:
Using \( \sin^2 x = \frac{b}{a+b} \) and \( \cos^2 x = \frac{a}{a+b} \):
\[ \sin^8 x = \left(\frac{b}{a+b}\right)^4 \quad \text{and} \quad \cos^8 x = \left(\frac{a}{a+b}\right)^4. \]

11. **Calculating the desired expression**:
\[ \frac{\sin^8 x}{b^3} + \frac{\cos^8 x}{a^3} = \frac{b^4}{(a+b)^4 b^3} + \frac{a^4}{(a+b)^4 a^3} = \frac{b}{(a+b)^4} + \frac{a}{(a+b)^4} = \frac{a+b}{(a+b)^4}. \]

12. **Final result**:
\[ = \frac{1}{(a+b)^3}. \]

### Final Answer:
\[ \frac{1}{(a+b)^3}. \]