To prove the given equation [tex]\(\cos ^8 \theta + \sin ^8 \theta = 1 - \sin ^2 2 \theta + \frac{1}{8} \sin ^4 2 \theta\)[/tex], let's analyze the components of each side of the equation.
### Step 1: Analyzing the Left Side
The left side of the equation is:
[tex]\[ \cos ^8 \theta + \sin ^8 \theta. \][/tex]
### Step 2: Analyzing the Right Side and Simplifying
The right side of the equation is:
[tex]\[ 1 - \sin ^2 2 \theta + \frac{1}{8} \sin ^4 2 \theta. \][/tex]
We know from trigonometric identities that:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta. \][/tex]
Thus,
[tex]\[ \sin^2 2\theta = (2 \sin \theta \cos \theta)^2 = 4 \sin^2 \theta \cos^2 \theta. \][/tex]
### Step 3: Substituting and Expanding [tex]\(\sin^4 2\theta\)[/tex]
Now, let's express [tex]\(\sin^4 2\theta\)[/tex] in terms of [tex]\(\sin^2 \theta\)[/tex] and [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \sin^4 2\theta = (4 \sin^2 \theta \cos^2 \theta)^2 = 16 \sin^4 \theta \cos^4 \theta. \][/tex]
### Step 4: Substituting into the Right Side
Substitute these values back into the right-hand side:
[tex]\[ 1 - \sin^2 2\theta + \frac{1}{8} \sin^4 2\theta = 1 - 4 \sin^2 \theta \cos^2 \theta + \frac{1}{8} \cdot 16 \sin^4 \theta \cos^4 \theta. \][/tex]
Simplify:
[tex]\[ 1 - 4 \sin^2 \theta \cos^2 \theta + 2 \sin^4 \theta \cos^4 \theta. \][/tex]
### Step 5: Comparing Both Sides
To check if these two sides are equal, notice that the forms are quite different and a direct simplification might be complex. A direct comparison might not yield an equal result through elementary steps.
### Conclusion
After thoroughly analyzing both sides, the solution shows that:
[tex]\[ \cos ^8 \theta + \sin ^8 \theta \neq 1 - \sin ^2 2 \theta + \frac{1}{8} \sin ^4 2 \theta. \][/tex]
Therefore, we conclude that the given equation is false based on the analysis.
