The possible value of `theta in [-pi, pi]` satisfying the equation `2(costheta + cos 2theta)+ (1+2costheta)sin 2theta = 2sintheta` are

Options

A. `-pi/2`
B. `-pi/3`
C. `pi/3`
D. `pi`

Answer

To solve the equation \( 2(\cos \theta + \cos 2\theta) + (1 + 2\cos \theta) \sin 2\theta = 2\sin \theta \), we will follow these steps:

### Step 1: Use Trigonometric Identities
We know that:
- \( \cos 2\theta = 2\cos^2 \theta - 1 \)
- \( \sin 2\theta = 2\sin \theta \cos \theta \)

Substituting these identities into the equation gives us:
\[ 2(\cos \theta + (2\cos^2 \theta - 1)) + (1 + 2\cos \theta)(2\sin \theta \cos \theta) = 2\sin \theta \]

### Step 2: Simplify the Equation
Expanding the equation:
\[ 2\cos \theta + 2(2\cos^2 \theta - 1) + (2\sin \theta \cos \theta + 4\sin \theta \cos^2 \theta) = 2\sin \theta \]
This simplifies to:
\[ 2\cos \theta + 4\cos^2 \theta - 2 + 2\sin \theta \cos \theta + 4\sin \theta \cos^2 \theta = 2\sin \theta \]

### Step 3: Rearranging Terms
Rearranging the equation gives us:
\[ 4\cos^2 \theta + 2\cos \theta + 2\sin \theta \cos \theta + 4\sin \theta \cos^2 \theta - 2 - 2\sin \theta = 0 \]
Combining like terms:
\[ 4\cos^2 \theta(1 + \sin \theta) + 2\cos \theta(1 + \sin \theta) - 2(1 + \sin \theta) = 0 \]

### Step 4: Factor the Equation
Factoring out \( (1 + \sin \theta) \):
\[ (1 + \sin \theta)(4\cos^2 \theta + 2\cos \theta - 2) = 0 \]

### Step 5: Solve Each Factor
1. **From \( 1 + \sin \theta = 0 \)**:
\[ \sin \theta = -1 \implies \theta = -\frac{\pi}{2} \]

2. **From \( 4\cos^2 \theta + 2\cos \theta - 2 = 0 \)**:
Using the quadratic formula \( \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = 4, b = 2, c = -2 \]
\[ \cos \theta = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 4 \cdot (-2)}}{2 \cdot 4} = \frac{-2 \pm \sqrt{4 + 32}}{8} = \frac{-2 \pm 6}{8} \]
This gives:
\[ \cos \theta = \frac{4}{8} = \frac{1}{2} \quad \text{or} \quad \cos \theta = \frac{-8}{8} = -1 \]

- **For \( \cos \theta = \frac{1}{2} \)**:
\[ \theta = \frac{\pi}{3}, -\frac{\pi}{3} \]

- **For \( \cos \theta = -1 \)**:
\[ \theta = \pi \]

### Step 6: Collect All Solutions
The possible values of \( \theta \) in the interval \([- \pi, \pi]\) are:
\[ \theta = -\frac{\pi}{2}, \frac{\pi}{3}, -\frac{\pi}{3}, \pi \]

### Final Answer
The possible values of \( \theta \) satisfying the equation are:
\[ \theta = -\frac{\pi}{2}, \frac{\pi}{3}, -\frac{\pi}{3}, \pi \]

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