Question

If $\alpha +\beta =\frac{\pi }{2}$, show that the maximum value of $\cos \alpha \cos \beta \text{is}\frac{1}{2}$

Answer 1

$\alpha +\beta =\frac{\pi }{2}\Rightarrow \beta =\frac{\pi }{2}-\alpha \Rightarrow \cos \beta =\cos (\frac{\pi }{2}-\alpha )=\sin \alpha$

$\Rightarrow y=\sin \alpha \cos \alpha =\frac{1}{2}\left(2\sin \alpha \cos \alpha \right)=\frac{1}{2}\sin 2\alpha \Rightarrow 1y1\le \frac{1}{2}$

$\{|\sin \theta |\le 1\}$