Question

Simplify the expression \( \sin \left( \frac{\pi}{3} + x \right) - \cos \left( \frac{\pi}{6} + x \right) \)

• A. \( -\cos x \)
• B. \( -\sin x \)
• C. \( \cos x \)
• D. \( \sin x \)

Hint:

When simplifying expressions with trigonometric functions, use sum or difference identities to break the terms down into simpler components.

Answer 1

The Correct Option is D

Step 1: Use trigonometric identities.
We apply the sum identities for sine and cosine: \[ \sin \left( \frac{\pi}{3} + x \right) = \sin \frac{\pi}{3} \cos x + \cos \frac{\pi}{3} \sin x \] and \[ \cos \left( \frac{\pi}{6} + x \right) = \cos \frac{\pi}{6} \cos x - \sin \frac{\pi}{6} \sin x \] Step 2: Substitute the values.
Using \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), \( \cos \frac{\pi}{3} = \frac{1}{2} \), \( \sin \frac{\pi}{6} = \frac{1}{2} \), and \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), we get: \[ \sin \left( \frac{\pi}{3} + x \right) - \cos \left( \frac{\pi}{6} + x \right) = \sin x \] Step 3: Conclusion.
The correct answer is (D) \( \sin x \).