If ` x = a cos ^(3) theta and y = b sin ^(3) theta`, then the value of `((x)/(a))^(2//3) + ((y)/(b))^(2//3)`

Options

A. 1
B. `-2`
C. 2
D. `-1`

Answer

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

1. **Given:**
\[ x = a \cos^3 \theta \quad \text{and} \quad y = b \sin^3 \theta \]

2. **We need to find:**
\[ \left( \frac{x}{a} \right)^{\frac{2}{3}} + \left( \frac{y}{b} \right)^{\frac{2}{3}} \]

3. **Substituting the values of \(x\) and \(y\):**
\[ \frac{x}{a} = \frac{a \cos^3 \theta}{a} = \cos^3 \theta \]
\[ \frac{y}{b} = \frac{b \sin^3 \theta}{b} = \sin^3 \theta \]

4. **Now, substituting these into the expression:**
\[ \left( \frac{x}{a} \right)^{\frac{2}{3}} + \left( \frac{y}{b} \right)^{\frac{2}{3}} = \left( \cos^3 \theta \right)^{\frac{2}{3}} + \left( \sin^3 \theta \right)^{\frac{2}{3}} \]

5. **Using the power of a power property:**
\[ \left( \cos^3 \theta \right)^{\frac{2}{3}} = \cos^{2} \theta \quad \text{and} \quad \left( \sin^3 \theta \right)^{\frac{2}{3}} = \sin^{2} \theta \]

6. **Now, we can rewrite the expression:**
\[ \cos^{2} \theta + \sin^{2} \theta \]

7. **Using the Pythagorean identity:**
\[ \cos^{2} \theta + \sin^{2} \theta = 1 \]

8. **Thus, the final answer is:**
\[ \left( \frac{x}{a} \right)^{\frac{2}{3}} + \left( \frac{y}{b} \right)^{\frac{2}{3}} = 1 \]

### Final Answer:
\[ \boxed{1} \]