Question

If \( \cos x - \sin x = 0 \), \( 0 \leq x \leq \pi \), then the value(s) of \( x \) is/are:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{3\pi}{4} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{5\pi}{4} \)
• E. \( \frac{3\pi}{2} \)

Hint:

For the equation \( \cos x = \sin x \), the solution is \( x = \frac{\pi}{4} \) for angles between \( 0 \) and \( \pi \).

Answer 1

The Correct Option is C

We are given the equation: \[ \cos x - \sin x = 0. \] This can be rewritten as: \[ \cos x = \sin x. \] Now, divide both sides of the equation by \( \cos x \) (assuming \( \cos x \neq 0 \)): \[ \frac{\sin x}{\cos x} = 1. \] The left-hand side of this equation is \( \tan x \), so we have: \[ \tan x = 1. \] The general solution to \( \tan x = 1 \) is: \[ x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}. \] We are given that \( 0 \leq x \leq \pi \), so we need to find the values of \( x \) within this interval. From the general solution, we get: \[ x = \frac{\pi}{4} \quad \text{(since \( n = 0 \))}. \] 
Thus, the only value of \( x \) in the interval \( 0 \leq x \leq \pi \) that satisfies \( \cos x = \sin x \) is \( x = \frac{\pi}{4} \). 
Thus, the correct answer is \( \boxed{\frac{\pi}{4}} \), corresponding to option (C).