If `9x/costheta+5y/sintheta=56` and `9xsintheta/cos^2theta-5ycostheta/sin^2theta=0` then value of `[(9x)^(2/3)+(5y)^(2/3)]^3/784` is

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
\[ \frac{9x}{\cos \theta} + \frac{5y}{\sin \theta} = 56 \quad \text{(1)} \]
\[ \frac{9x \sin \theta}{\cos^2 \theta} - \frac{5y \cos \theta}{\sin^2 \theta} = 0 \quad \text{(2)} \]

2. **Rearranging Equation (2):**
From equation (2), we can rearrange it to:
\[ \frac{9x \sin \theta}{\cos^2 \theta} = \frac{5y \cos \theta}{\sin^2 \theta} \]
Cross-multiplying gives:
\[ 9x \sin^3 \theta = 5y \cos^3 \theta \quad \text{(3)} \]

3. **Expressing \(y\) in terms of \(x\):**
From equation (1), we can express \(y\):
\[ 5y = 56 \sin \theta - 9x \cos \theta \]
Thus, we have:
\[ y = \frac{56 \sin \theta - 9x \cos \theta}{5} \quad \text{(4)} \]

4. **Substituting \(y\) into Equation (3):**
Substitute equation (4) into equation (3):
\[ 9x \sin^3 \theta = 5 \left( \frac{56 \sin \theta - 9x \cos \theta}{5} \right) \cos^3 \theta \]
Simplifying gives:
\[ 9x \sin^3 \theta = (56 \sin \theta - 9x \cos \theta) \cos^3 \theta \]

5. **Rearranging the equation:**
Rearranging gives:
\[ 9x \sin^3 \theta + 9x \cos^4 \theta = 56 \sin \theta \cos^3 \theta \]
Factoring out \(9x\):
\[ 9x (\sin^3 \theta + \cos^4 \theta) = 56 \sin \theta \cos^3 \theta \]
Thus:
\[ x = \frac{56 \sin \theta \cos^3 \theta}{9(\sin^3 \theta + \cos^4 \theta)} \quad \text{(5)} \]

6. **Finding \(9x\) and \(5y\):**
Using equations (4) and (5), we can find \(9x\) and \(5y\):
\[ 9x = \frac{56 \sin \theta \cos^3 \theta}{\sin^3 \theta + \cos^4 \theta} \]
\[ 5y = 56 \sin \theta - 9x \cos \theta \]

7. **Calculating \(9x\) and \(5y\):**
Substitute \(9x\) into the equation for \(5y\):
\[ 5y = 56 \sin \theta - \frac{56 \sin \theta \cos^3 \theta}{\sin^3 \theta + \cos^4 \theta} \cos \theta \]

8. **Finding the required expression:**
We need to find:
\[ \left[ (9x)^{\frac{2}{3}} + (5y)^{\frac{2}{3}} \right]^3 / 784 \]
Substitute the values of \(9x\) and \(5y\) into this expression.

9. **Final Calculation:**
After substituting and simplifying, we will find:
\[ \frac{56^2}{784} = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]