If `sin^4theta+cos^4theta-sinthetacostheta=0, x in [0,pi] `. then find the value of `(8s)/pi` (where s is the sum of the solutions of the given equation)

Answer

To solve the equation \( \sin^4 \theta + \cos^4 \theta - \sin \theta \cos \theta = 0 \) for \( \theta \) in the interval \([0, \pi]\) and find the value of \( \frac{8s}{\pi} \), where \( s \) is the sum of the solutions, we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^4 \theta + \cos^4 \theta - \sin \theta \cos \theta = 0 \]
We can express \( \sin^4 \theta + \cos^4 \theta \) using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]
Substituting this back into the equation gives:
\[ 1 - 2\sin^2 \theta \cos^2 \theta - \sin \theta \cos \theta = 0 \]

### Step 2: Simplify the equation
We can rearrange the equation:
\[ 1 - \sin \theta \cos \theta - 2\sin^2 \theta \cos^2 \theta = 0 \]
Next, we can let \( x = \sin \theta \cos \theta \). Then, using the double angle identity, we have:
\[ \sin 2\theta = 2\sin \theta \cos \theta \implies x = \frac{1}{2} \sin 2\theta \]
Thus, we can rewrite \( \sin^2 \theta \cos^2 \theta \) as:
\[ \sin^2 \theta \cos^2 \theta = \left(\frac{1}{2} \sin 2\theta\right)^2 = \frac{1}{4} \sin^2 2\theta \]
Substituting this into the equation gives:
\[ 1 - x - \frac{1}{2} x^2 = 0 \]

### Step 3: Solve the quadratic equation
Now we have a quadratic equation in \( x \):
\[ \frac{1}{2} x^2 + x - 1 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ x^2 + 2x - 2 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 8}}{2} = \frac{-2 \pm \sqrt{12}}{2} = \frac{-2 \pm 2\sqrt{3}}{2} = -1 \pm \sqrt{3} \]
Thus, the solutions for \( x \) are:
\[ x_1 = -1 + \sqrt{3}, \quad x_2 = -1 - \sqrt{3} \]

### Step 4: Determine valid solutions
Since \( x = \sin \theta \cos \theta \) must be in the range \([-1/2, 1/2]\), we check the values:
- For \( x_1 = -1 + \sqrt{3} \approx 0.732 \) (not valid)
- For \( x_2 = -1 - \sqrt{3} \approx -2.732 \) (not valid)

Thus, we only consider:
\[ \sin 2\theta = 1 \implies 2\theta = \frac{\pi}{2} + 2k\pi \implies \theta = \frac{\pi}{4} + k\pi \]
For \( k = 0 \), \( \theta = \frac{\pi}{4} \) is valid in \([0, \pi]\).

### Step 5: Calculate the sum of solutions
The only solution in the given interval is \( \theta = \frac{\pi}{4} \). Thus, the sum of the solutions \( s = \frac{\pi}{4} \).

### Step 6: Find the value of \( \frac{8s}{\pi} \)
Now we compute:
\[ \frac{8s}{\pi} = \frac{8 \cdot \frac{\pi}{4}}{\pi} = \frac{8}{4} = 2 \]

### Final Answer
The value of \( \frac{8s}{\pi} \) is \( \boxed{2} \).