Question

If \( \sin x = \sin y \) and \( \cos x = \cos y \), then the value of \( x - y \) is:

• A. \(\frac{\pi}{4}\)
• B. \( n\pi/2 \)
• C. \( n\pi \)
• D. \( 2n\pi \)

Hint:

When dealing with trigonometric equations, remember the periodic nature of sine and cosine functions.

Answer 1

The Correct Option is D

We are given the equations \( \sin x = \sin y \) and \( \cos x = \cos y \), and we are tasked with finding the value of \( x - y \). 
Step 1: Use the identities for sine and cosine.
We know that: \[ \sin x = \sin y \quad \Rightarrow \quad x = y + 2n\pi \, \text{or} \, x = \pi - y + 2n\pi \quad \text{(for some integer } n\text{)}. \] Also, from \( \cos x = \cos y \), we have: \[ x = y + 2n\pi \quad \text{or} \quad x = -y + 2n\pi \quad \text{(for some integer } n\text{)}. \] Step 2: Analyze the possible solutions.
From both conditions, the only consistent solution is: \[ x - y = 2n\pi \quad \text{(for some integer } n\text{)}. \] Thus, the correct answer is: \[ \boxed{2n\pi}. \]