Solve `5 cos 2 theta+2 "cos"^(2) theta/2 +1=0, -pi lt theta lt pi`.

Explanation

To solve the equation \( 5 \cos 2\theta + 2 \cos^2 \frac{\theta}{2} + 1 = 0 \) for \( -\pi < \theta < \pi \), we can follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \cos 2\theta = 2 \cos^2 \theta - 1 \). Substituting this into the equation gives:
\[ 5(2 \cos^2 \theta - 1) + 2 \cos^2 \frac{\theta}{2} + 1 = 0 \]

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