Evaluate \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\)

To evaluate \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\), we can use the formula for the cosine of the sum of two angles, which states that \[\cos(a+b) = \cos a \cos b - \sin a \sin b.\] Using this formula, we can simplify \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\) as follows: \[\cos (\frac{\pi}{2} + \frac{\pi}{3}) = \cos \frac{\pi}{2} \cos \frac{\pi}{3} - \sin \frac{\pi}{2} \sin \frac{\pi}{3}\] Recall that \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\), and we also know that \(\cos \frac{\pi}{3} = \frac{1}{2}\) and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\) from the unit circle. Substituting these values into the equation above, we get: \[\cos (\frac{\pi}{2} + \frac{\pi}{3}) = 0 \cdot \frac{1}{2} - 1 \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}\] Therefore, the value of \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\) is \(\frac{-\sqrt{3}}{2}\)