\[- 1 \leq \frac{\sqrt{3}}{2}\sin a - \frac{1}{2}\cos a \leq 1\]
\[\frac{\sqrt{3}}{2}\sin a - \frac{1}{2}\cos a =\]
\[= \cos\frac{\pi}{6}\sin a - \sin\frac{\pi}{6}\cos a =\]
\[= \sin\left( a - \frac{\pi}{6} \right)\]
\[- 1 \leq \sin\left( a - \frac{\pi}{6} \right) \leq 1 \Longrightarrow\]
\[\Longrightarrow - 1 \leq \frac{\sqrt{3}}{2}\sin a - \frac{1}{2}\cos a \leq 1.\]