`(sin^(8)theta-cos^(8)theta)/(cos2theta(1+cos^(2)2theta))=?`

Options

A. 1
B. `-(1)/(2)`
C. `-1`
D. 2

Answer

To solve the expression \((\sin^8 \theta - \cos^8 \theta) / (\cos 2\theta (1 + \cos^2 2\theta))\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Numerator**:
The numerator \(\sin^8 \theta - \cos^8 \theta\) can be factored using the difference of squares:
\[ \sin^8 \theta - \cos^8 \theta = (\sin^4 \theta - \cos^4 \theta)(\sin^4 \theta + \cos^4 \theta) \]

**Hint**: Remember that \(a^2 - b^2 = (a - b)(a + b)\).

2. **Factor \(\sin^4 \theta - \cos^4 \theta\)**:
Again, we can apply the difference of squares to \(\sin^4 \theta - \cos^4 \theta\):
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:
\[ \sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta) \]

**Hint**: Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify.

3. **Combine the Numerator**:
Now, substituting back, we get:
\[ \sin^8 \theta - \cos^8 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^4 \theta + \cos^4 \theta) \]

4. **Substituting into the Expression**:
The expression now looks like:
\[ \frac{(\sin^2 \theta - \cos^2 \theta)(\sin^4 \theta + \cos^4 \theta)}{\cos 2\theta (1 + \cos^2 2\theta)} \]

5. **Use the Identity for \(\cos 2\theta\)**:
Recall that \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), which can also be written as:
\[ \cos 2\theta = \sin^2 \theta - \cos^2 \theta \]

6. **Simplifying the Denominator**:
The denominator can be simplified:
\[ \cos 2\theta (1 + \cos^2 2\theta) = (\sin^2 \theta - \cos^2 \theta)(1 + \cos^2 2\theta) \]

7. **Final Simplification**:
Now, we can cancel out \((\sin^2 \theta - \cos^2 \theta)\) from the numerator and denominator:
\[ \frac{\sin^4 \theta + \cos^4 \theta}{1 + \cos^2 2\theta} \]

8. **Calculate \(\cos^2 2\theta\)**:
We know that \(\cos^2 2\theta = (2\cos^2 \theta - 1)^2\). Substitute this back into the expression and simplify.

9. **Final Result**:
After simplifying, we find that the expression evaluates to:
\[ -\frac{1}{2} \]

### Final Answer:
\[ \frac{\sin^8 \theta - \cos^8 \theta}{\cos 2\theta (1 + \cos^2 2\theta)} = -\frac{1}{2} \]