To simplify the given expression $\cos\left(x + \frac{\pi}{6}\right) + \sin\left(x - \frac{\pi}{3}\right)$, let us break it down step by step.

Step 1: Expand the trigonometric terms using angle addition and subtraction formulas.

1. Cosine addition formula: $\cos\left(x + \frac{\pi}{6}\right) = \cos x \cos\frac{\pi}{6} - \sin x \sin\frac{\pi}{6}$
2. Sine subtraction formula: $\sin\left(x - \frac{\pi}{3}\right) = \sin x \cos\frac{\pi}{3} - \cos x \sin\frac{\pi}{3}$

Step 2: Substitute the exact values of trigonometric functions for $\frac{\pi}{6}$ and $\frac{\pi}{3}$:

• $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$,
• $\sin\frac{\pi}{6} = \frac{1}{2}$,
• $\cos\frac{\pi}{3} = \frac{1}{2}$,
• $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$.

Substitute these into the formulas: $\cos\left(x + \frac{\pi}{6}\right) = \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}$ $\sin\left(x - \frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}$

Step 3: Combine the terms.

Add the two expressions: