If `(cos^(4)alpha)/(cos^(2)beta)+(sin^(4)beta)/(sin^(2)alpha)=1`, then `(cos^(6)beta)/(cos^(4)alpha)+(sin^(6)alpha)/(sin^(4)beta)=`?

Options

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

Answer

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

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

We need to find the value of:

\[ \frac{\cos^6 \beta}{\cos^4 \alpha} + \frac{\sin^6 \alpha}{\sin^4 \beta} \]

### Step 1: Substitute values for \(\alpha\) and \(\beta\)

Since there are no specific conditions given for \(\alpha\) and \(\beta\), we can assume values for simplicity. Let's take:

\[ \alpha = \beta = 45^\circ \]

### Step 2: Calculate \(\cos^2 \alpha\) and \(\sin^2 \alpha\)

Using the values of trigonometric functions at \(45^\circ\):

\[ \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \]

Thus:

\[ \cos^2 45^\circ = \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 3: Verify the initial condition

Substituting \(\alpha\) and \(\beta\) into the initial equation:

\[ \frac{\left(\frac{1}{2}\right)^2}{\frac{1}{2}} + \frac{\left(\frac{1}{2}\right)^2}{\frac{1}{2}} = \frac{\frac{1}{4}}{\frac{1}{2}} + \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} + \frac{1}{2} = 1 \]

This confirms that our assumption satisfies the initial condition.

### Step 4: Calculate the required expression

Now substituting \(\alpha\) and \(\beta\) into the expression we need to evaluate:

\[ \frac{\cos^6 45^\circ}{\cos^4 45^\circ} + \frac{\sin^6 45^\circ}{\sin^4 45^\circ} \]

Calculating each term:

1. **First term**:
\[ \frac{\cos^6 45^\circ}{\cos^4 45^\circ} = \frac{\left(\frac{1}{\sqrt{2}}\right)^6}{\left(\frac{1}{\sqrt{2}}\right)^4} = \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{8} \cdot 4 = \frac{1}{2} \]

2. **Second term**:
\[ \frac{\sin^6 45^\circ}{\sin^4 45^\circ} = \frac{\left(\frac{1}{\sqrt{2}}\right)^6}{\left(\frac{1}{\sqrt{2}}\right)^4} = \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{8} \cdot 4 = \frac{1}{2} \]

### Step 5: Add the two terms

Now, adding both terms:

\[ \frac{1}{2} + \frac{1}{2} = 1 \]

### Conclusion

Thus, the value of

\[ \frac{\cos^6 \beta}{\cos^4 \alpha} + \frac{\sin^6 \alpha}{\sin^4 \beta} = 1 \]

### Final Answer

The answer is:

\[ \boxed{1} \]