Question

If \( \cos \theta + \sin \theta = \sqrt{2} \), then \( \cos \theta - \sin \theta \) is equal to:

• A. 0
• B. \(-\frac{1}{2}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{1}{4}\)
• E. 1

Hint:

When given \( \cos \theta + \sin \theta \), squaring the equation helps eliminate the terms and leads to finding the value of \( \cos \theta - \sin \theta \).

Answer 1

The Correct Option is A

Step 1: We are given \( \cos \theta + \sin \theta = \sqrt{2} \). We square both sides of the equation: \[ (\cos \theta + \sin \theta)^2 = (\sqrt{2})^2 \] Expanding the left side: \[ \cos^2 \theta + 2 \cos \theta \sin \theta + \sin^2 \theta = 2 \] Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ 1 + 2 \cos \theta \sin \theta = 2 \quad \Rightarrow \quad 2 \cos \theta \sin \theta = 1 \] Thus, \( \cos \theta \sin \theta = \frac{1}{2} \). 
Step 2: Now we calculate \( \cos \theta - \sin \theta \) by squaring: \[ (\cos \theta - \sin \theta)^2 = \cos^2 \theta - 2 \cos \theta \sin \theta + \sin^2 \theta \] Substituting the known values: \[ 1 - 2 \times \frac{1}{2} = 1 - 1 = 0 \] Thus, \( \cos \theta - \sin \theta = 0 \).