If `(cos^(4)alpha)/(cos^(2)beta)+(sin^(4)alpha)/(sin^(2)beta)=1` then the value of `(cos^(4)beta)/(cos^(2)alpha)+(sin^(4)beta)/(sin^(2)alpha)` is

Options

A. 4
B. 0
C. `(1)/(8)`
D. 1

Answer

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

\[ \frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1 \]

We need to find the value of:

\[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} \]

### Step 1: Rewrite the Given Equation
We can rewrite the left-hand side of the given equation by taking a common denominator:

\[ \frac{\cos^4 \alpha \sin^2 \beta + \sin^4 \alpha \cos^2 \beta}{\cos^2 \beta \sin^2 \beta} = 1 \]

This implies:

\[ \cos^4 \alpha \sin^2 \beta + \sin^4 \alpha \cos^2 \beta = \cos^2 \beta \sin^2 \beta \]

### Step 2: Express the Required Expression
Now, we express the required expression:

\[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} \]

Taking a common denominator, we have:

\[ \frac{\cos^4 \beta \sin^2 \alpha + \sin^4 \beta \cos^2 \alpha}{\cos^2 \alpha \sin^2 \alpha} \]

### Step 3: Relate the Two Expressions
Notice that we can relate the two expressions. We know from the first equation that:

\[ \cos^4 \alpha \sin^2 \beta + \sin^4 \alpha \cos^2 \beta = \cos^2 \beta \sin^2 \beta \]

Using the identity \( \sin^2 \beta = 1 - \cos^2 \beta \), we can substitute this into our expression.

### Step 4: Substitute and Simplify
Substituting \( \sin^2 \beta \) into the equation gives us:

\[ \cos^4 \alpha (1 - \cos^2 \beta) + \sin^4 \alpha \cos^2 \beta = \cos^2 \beta (1 - \cos^2 \beta) \]

### Step 5: Analyze the Result
This leads us to see that both expressions are symmetric in nature. If we assume \( \cos^2 \alpha = \cos^2 \beta \) and \( \sin^2 \alpha = \sin^2 \beta \), we can conclude that:

\[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} = 1 \]

### Final Result
Thus, the value of the expression we need to find is:

\[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} = 1 \]

### Conclusion
The answer is:

\[ \boxed{1} \]