Question

Simplify the following expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} \] The simplified form is:

• A. \( \frac{\tan 6x}{\sin 6x} \)
• B. \( \frac{\tan 6x}{\cos 6x} \)
• C. \( \sin 6x \)
• D. \( \cos 6x \)
• E. \( \tan 6x \)

Hint:

When simplifying trigonometric expressions, apply sum-to-product identities to break down complex terms. This can often lead to a simpler form for easier calculation or understanding.

Answer 1

Answer: (E)

We are tasked with simplifying the expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x}. \] Step 1: Use the sum-to-product identities for sine and cosine. 
The sum-to-product identity for sine is: \[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right), \] and for cosine: \[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right). \] Step 2: Apply the sum-to-product identities to the given expression: 
- For \( \sin 7x + \sin 5x \), we have: \[ \sin 7x + \sin 5x = 2 \sin\left(\frac{7x + 5x}{2}\right) \cos\left(\frac{7x - 5x}{2}\right) = 2 \sin(6x) \cos(x). \] - For \( \cos 7x + \cos 5x \), we have: \[ \cos 7x + \cos 5x = 2 \cos\left(\frac{7x + 5x}{2}\right) \cos\left(\frac{7x - 5x}{2}\right) = 2 \cos(6x) \cos(x). \] Step 3: Substitute these into the original expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} = \frac{2 \sin(6x) \cos(x)}{2 \cos(6x) \cos(x)}. \] Simplify by canceling out \( 2 \cos(x) \): \[ \frac{\sin(6x)}{\cos(6x)} = \tan(6x). \] 
Thus, the simplified form of the expression is \( \tan 6x \).
Therefore, the correct answer is option (E).