Connect with experts and get insightful answers to your questions on IDNLearn.com. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.

Prove: [tex]$\cos^8 \theta + \sin^8 \theta = 1 - \sin^2 2\theta + \frac{1}{8} \sin^4 2\theta$[/tex]

Sagot :

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.