The set of values of `lambda` for which the equation `"sin"^(4) x + "cos"^(4) x =lambda` has a solution, is

Options

A. `(0, 1)`
B. `(1, 3//2)`
C. `[-1, 1]`
D. `[1//2, 1]`

Explanation

To solve the equation \( \sin^4 x + \cos^4 x = \lambda \) and find the set of values of \( \lambda \) for which this equation has a solution, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^4 x + \cos^4 x = \lambda \]
This can be rewritten using the identity for the sum of squares:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting this into our equation gives:
\[ \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x \]

### Step 3: Express \( \sin^2 x \cos^2 x \) in terms of \( \sin^2 2x \)
Using the double angle identity, we have:
\[ \sin 2x = 2 \sin x \cos x \]
Squaring both sides, we get:
\[ \sin^2 2x = 4\sin^2 x \cos^2 x \]
Thus, we can express \( \sin^2 x \cos^2 x \) as:
\[ \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x \]

### Step 4: Substitute back into the equation
Substituting this back into our equation gives:
\[ \sin^4 x + \cos^4 x = 1 - 2 \left(\frac{1}{4} \sin^2 2x\right) = 1 - \frac{1}{2} \sin^2 2x \]
So we have:
\[ \sin^4 x + \cos^4 x = 1 - \frac{1}{2} \sin^2 2x = \lambda \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ \frac{1}{2} \sin^2 2x = 1 - \lambda \]
Thus,
\[ \sin^2 2x = 2(1 - \lambda) \]

### Step 6: Determine the range of \( \sin^2 2x \)
Since \( \sin^2 2x \) must lie between 0 and 1, we have:
\[ 0 \leq 2(1 - \lambda) \leq 1 \]

### Step 7: Solve the inequalities
From \( 0 \leq 2(1 - \lambda) \):
\[ 1 - \lambda \geq 0 \implies \lambda \leq 1 \]

From \( 2(1 - \lambda) \leq 1 \):
\[ 1 - \lambda \leq \frac{1}{2} \implies \lambda \geq \frac{1}{2} \]

### Conclusion
Combining these inequalities, we find:
\[ \frac{1}{2} \leq \lambda \leq 1 \]
Thus, the set of values of \( \lambda \) for which the equation has a solution is:
\[ \lambda \in \left[\frac{1}{2}, 1\right] \]