Prove that: `1+cos^2 2x=2(cos^4x+sin^4x)`

Explanation

To prove the equation \(1 + \cos^2(2x) = 2(\cos^4(x) + \sin^4(x))\), we will follow these steps:

### Step 1: Rewrite \(\cos(2x)\)
We know that:
\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]
Thus, we can express \(\cos^2(2x)\) as:
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