## Precalculus (6th Edition) Blitzer

Let us consider the left side of the given expression: $\cos \frac{\pi }{2}\cos \frac{\pi }{3}$ Now, put the values of the trigonometric functions to find the exact value as shown below: \begin{align} & \cos \frac{\pi }{2}\cos \frac{\pi }{3}=0\cdot \frac{1}{2} \\ & =0 \end{align} Also, consider the right side of the given expression: $\frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)+\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]$ Then, put the values of the trigonometric functions to find the exact value as shown below: \begin{align} & \frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)+\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]=\frac{1}{2}\left[ \cos \left( \frac{3\pi -2\pi }{6} \right)+\cos \left( \frac{3\pi +2\pi }{6} \right) \right] \\ & =\frac{1}{2}\left[ \cos \left( \frac{\pi }{6} \right)+\cos \left( \frac{5\pi }{6} \right) \right] \\ & =\frac{1}{2}\cdot \left[ \frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2} \right] \\ & =0 \end{align} Thus, the left side of the expression is equal to the right side, which is $\cos \frac{\pi }{2}\cos \frac{\pi }{3}=\frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)-\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]$.