Number of roots of the equation `(3+cos"x")^2=4-2sin^8x ,x in [0,5pi]a r e` _________

Explanation

To find the number of roots of the equation \((3 + \cos x)^2 = 4 - 2 \sin^8 x\) for \(x\) in the interval \([0, 5\pi]\), we will analyze both sides of the equation step by step.

### Step 1: Analyze the Left-Hand Side (LHS)
The LHS of the equation is given by:
\[ (3 + \cos x)^2 \]
The cosine function, \(\cos x\), varies between -1 and 1. Therefore, we can find the minimum and maximum values of \(3 + \cos x\):
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