The number of solutions of equation `6cos2theta+2cos^2(theta/2)+2sin^2theta=0, ` `-πltθltπ` is 3 (b) 4 (c) 5 (d) 6

Explanation

To solve the equation \( 6 \cos(2\theta) + 2 \cos^2\left(\frac{\theta}{2}\right) + 2 \sin^2(\theta) = 0 \) for the number of solutions in the interval \( -\pi < \theta < \pi \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ 6 \cos(2\theta) + 2 \cos^2\left(\frac{\theta}{2}\right) + 2 \sin^2(\theta) = 0 \]

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