If `x=acos^(3)theta,y=bsin^(3)theta` then `((x)/(a))^((2)/(3))+((y)/(b))^((2)/(3))=?`

Options

A. 1
B. 0
C. 2
D. 4

Answer

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

1. \( x = a \cos^3 \theta \)
2. \( y = b \sin^3 \theta \)

We need to find the value of:

\[ \left( \frac{x}{a} \right)^{\frac{2}{3}} + \left( \frac{y}{b} \right)^{\frac{2}{3}} \]

### Step 1: Simplify \( \frac{x}{a} \)

From the first equation, we can express \( \frac{x}{a} \):

\[ \frac{x}{a} = \frac{a \cos^3 \theta}{a} = \cos^3 \theta \]

### Step 2: Simplify \( \frac{y}{b} \)

Now, we simplify \( \frac{y}{b} \):

\[ \frac{y}{b} = \frac{b \sin^3 \theta}{b} = \sin^3 \theta \]

### Step 3: Substitute into the expression

Now we substitute these values into the expression we need to evaluate:

\[ \left( \frac{x}{a} \right)^{\frac{2}{3}} + \left( \frac{y}{b} \right)^{\frac{2}{3}} = \left( \cos^3 \theta \right)^{\frac{2}{3}} + \left( \sin^3 \theta \right)^{\frac{2}{3}} \]

### Step 4: Apply the exponent

Using the property of exponents, we can simplify further:

\[ \left( \cos^3 \theta \right)^{\frac{2}{3}} = \cos^2 \theta \]
\[ \left( \sin^3 \theta \right)^{\frac{2}{3}} = \sin^2 \theta \]

### Step 5: Combine the results

Now we can combine these results:

\[ \cos^2 \theta + \sin^2 \theta \]

### Step 6: Use the Pythagorean identity

Using the Pythagorean identity, we know that:

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

### Final Result

Thus, the value of the expression is:

\[ \boxed{1} \]

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