If `(cos^(4)alpha)/(cos^(2)beta)+(sin^(4)alpha)/(sin^(2)beta)=1` then `(cos^(4)beta)/(cos^(2)alpha)+(sin^(4)beta)/(sin^(2)alpha)=?`

Options

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

Answer

To solve the given problem step by step, we start with the 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: Substitute \(\alpha = \beta\)

To simplify the problem, we can assume \(\alpha = \beta\). This is a common technique in trigonometric identities, especially when the problem is symmetric.

### Step 2: Rewrite the given equation

Substituting \(\alpha = \beta\) into the original equation gives us:

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

### Step 3: Simplify the equation

This simplifies to:

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

### Step 4: Use the Pythagorean identity

We know from the Pythagorean identity that:

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

This confirms that our assumption holds true.

### Step 5: Find the desired expression

Now, we need to evaluate:

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

Substituting \(\beta = \alpha\) gives:

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

### Step 6: Simplify the new expression

This simplifies to:

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

### Step 7: Use the Pythagorean identity again

Again, using the identity:

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

### Conclusion

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

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