`2 sin^2 x + sin^2 2 x =2, - pi lt x lt pi`, then x =

Options

A. `pm pi//2`
B. `pm pi//4`
C. `pm3 pi//4`
D. none

Answer

To solve the equation \(2 \sin^2 x + \sin^2 2x = 2\) for \(-\pi < x < \pi\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 2 \sin^2 x + \sin^2 2x = 2 \]

### Step 2: Use the double angle identity
Recall that \(\sin 2x = 2 \sin x \cos x\). Therefore,
\[ \sin^2 2x = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x \]

### Step 3: Substitute into the equation
Substituting \(\sin^2 2x\) into the original equation gives:
\[ 2 \sin^2 x + 4 \sin^2 x \cos^2 x = 2 \]

### Step 4: Factor out \(\sin^2 x\)
Let \(y = \sin^2 x\). Then we can rewrite the equation as:
\[ 2y + 4y(1 - y) = 2 \]
This simplifies to:
\[ 2y + 4y - 4y^2 = 2 \]
\[ 6y - 4y^2 = 2 \]

### Step 5: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ 4y^2 - 6y + 2 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 4\), \(b = -6\), and \(c = 2\).
\[ y = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 4 \cdot 2}}{2 \cdot 4} \]
\[ y = \frac{6 \pm \sqrt{36 - 32}}{8} \]
\[ y = \frac{6 \pm 2}{8} \]
Calculating the two possible values:
1. \(y = \frac{8}{8} = 1\)
2. \(y = \frac{4}{8} = \frac{1}{2}\)

### Step 7: Find values of \(x\)
Now we have two cases for \(\sin^2 x\):
1. \(\sin^2 x = 1\) implies \(\sin x = \pm 1\) which gives \(x = \pm \frac{\pi}{2}\).
2. \(\sin^2 x = \frac{1}{2}\) implies \(\sin x = \pm \frac{1}{\sqrt{2}}\) which gives \(x = \pm \frac{\pi}{4}\).

### Step 8: Compile all solutions
The solutions for \(x\) in the interval \(-\pi < x < \pi\) are:
\[ x = \pm \frac{\pi}{4}, \pm \frac{\pi}{2} \]

### Step 9: Check for any additional solutions
We also need to check if there are any solutions from the double angle identity:
- For \(2x = \frac{\pi}{2} + n\pi\) and \(2x = \pi + n\pi\), we find that:
- \(x = \frac{\pi}{4}\), \(-\frac{\pi}{4}\), \(\frac{\pi}{2}\), and \(-\frac{\pi}{2}\) are valid solutions.

### Final Solutions
Thus, the final solutions are:
\[ x = \frac{\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{2}, -\frac{\pi}{2} \]