https://file.aiquickdraw.com/mathbot/file/1729743627799hck8p5m8.jpg

a.
-tan 6x


b.
cot 6x


c.
-tan x


d.
cot x


Which one?

We are tasked with simplifying the expression given as:

\[
\frac{\cos(5x) - \cos(7x)}{\sin(5x) - \sin(7x)}
\]

We can use trigonometric identities to simplify the numerator and the denominator.

### Step 1: Apply the trigonometric identity for the difference of cosines:

\[
\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)
\]

For the numerator \(\cos(5x) - \cos(7x)\):

\[
\cos(5x) - \cos(7x) = -2 \sin\left(\frac{5x + 7x}{2}\right)\sin\left(\frac{5x - 7x}{2}\right)
\]
\[
= -2 \sin(6x)\sin(-x) = 2 \sin(6x)\sin(x)
\]
(Note that \(\sin(-x) = -\sin(x)\), so the negative sign cancels out.)

### Step 2: Apply the trigonometric identity for the difference of sines:

\[
\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)
\]

For the denominator \(\sin(5x) - \sin(7x)\):

\[
\sin(5x) - \sin(7x) = 2 \cos\left(\frac{5x + 7x}{2}\right)\sin\left(\frac{5x - 7x}{2}\right)
\]
\[
= 2 \cos(6x)\sin(-x) = -2 \cos(6x)\sin(x)
\]
(Again, using \(\sin(-x) = -\sin(x)\), so we include the negative sign here.)

### Step 3: Simplify the expression

The simplified expression becomes:

\[
\frac{2 \sin(6x) \sin(x)}{-2 \cos(6x) \sin(x)}
\]

Canceling the common terms \(2 \sin(x)\) in the numerator and denominator:

\[
= -\frac{\sin(6x)}{\cos(6x)}
\]

We recognize this as:

\[
= -\tan(6x)
\]

### Final Answer:

The correct answer is **(a) \(-\tan(6x)\)**.

Would you like more details on any step, or do you have further questions?

Here are 5 related questions to expand this topic:
1. How would you simplify \(\frac{\cos 3x - \cos 9x}{\sin 3x - \sin 9x}\)?
2. What is the general form of the difference of cosines identity?
3. How does the simplification change if the angles were \(4x\) and \(6x\) instead of \(5x\) and \(7x\)?
4. What is the relationship between tangent and sine/cosine?
5. Could you apply a similar method to simplify expressions involving tangent functions?

**Tip:** Always look for common factors to cancel in trigonometric identities; it can greatly simplify the problem.

Simplify the expression (cos 5x - cos 7x) / (sin 5x - sin 7x)

Learn how to simplify the trigonometric expression (cos 5x - cos 7x) / (sin 5x - sin 7x) by applying key identities. This problem covers advanced trigonometric simplifications and the use of tangent functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation