Let's simplify the given trigonometric expression:

The expression is:

$\frac{2 \sin(4\alpha) \cos(-3\alpha) + 2 \sin(4\alpha) \cos(-\alpha)}{2 \cos(4\alpha) \cos(-3\alpha) + 2 \cos(4\alpha) \cos(-\alpha)}$

Step 1: Factor out the common terms

Both the numerator and denominator have a common factor of $2$, which can be canceled:

[ \frac{2 \left[\sin(4\alpha) \cos(-3\alpha) + \sin(4\alpha) \cos(-\alpha)\right]}{2 \left[\cos(4\alpha) \cos(-3\alpha) + \cos(4\alpha) \cos(-\alpha)\right]} ]

Cancel the $2$'s:

$\frac{\sin(4\alpha) \cos(-3\alpha) + \sin(4\alpha) \cos(-\alpha)}{\cos(4\alpha) \cos(-3\alpha) + \cos(4\alpha) \cos(-\alpha)}$

Step 2: Simplify using trigonometric properties

The cosine function is even, meaning $\cos(-x) = \cos(x)$, and the sine function is odd, meaning $\sin(-x) = -\sin(x)$. Apply this to the expression:

• $\cos(-3\alpha) = \cos(3\alpha)$
• $\cos(-\alpha) = \cos(\alpha)$

Thus, the expression becomes:

$\frac{\sin(4\alpha)\cos(3\alpha) + \sin(4\alpha)\cos(\alpha)}{\cos(4\alpha)\cos(3\alpha) + \cos(4\alpha)\cos(\alpha)}.$

Step 3: Factor out common terms in both numerator and denominator

In the numerator, factor out $\sin(4\alpha)$, and in the denominator, factor out $\cos(4\alpha)$:

[ \frac{\sin(4\alpha)\left[\cos(3\alpha) + \cos(\alpha)\right]}{\cos(4\alpha)\left[\cos(3\alpha) + \cos(\alpha)\right]}. ]

Step 4: Cancel the common term

If $\cos(3\alpha) + \cos(\alpha) \neq 0$, then we can cancel $\left[\cos(3\alpha) + \cos(\alpha)\right]$ from both numerator and denominator:

$\frac{\sin(4\alpha)}{\cos(4\alpha)}.$

Step 5: Final Simplification

The resulting expression simplifies to:

$\tan(4\alpha).$

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Final Answer:

The simplified expression is:

$\tan(4\alpha).$

Let me know if you would like further clarification or a breakdown of any steps!

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Related Questions:

1. What are the properties of sine and cosine for negative angles?
2. How do you simplify expressions involving common factors in trigonometry?
3. What is the tangent function in terms of sine and cosine?
4. Can the tangent function be expressed using double or multiple angles?
5. How do you prove trigonometric identities step by step?

Tip:

When simplifying trigonometric expressions, always check for common factors and use trigonometric properties such as the evenness of cosine and the oddness of sine.