Question

If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].

 

 

Answer 1

\[\frac{\pi}{2} = 90^\circ\]

\[\text{ Let }x = \cos \alpha \cos \beta\]

\[ \Rightarrow x = \frac{1}{2}\left[ 2\cos \alpha \cos \beta \right]\]

\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha + \beta \right) + \cos \left( \alpha - \beta \right) \right]\]

\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha - \beta \right) + \cos 90^\circ \right]\]

\[ \Rightarrow x = \frac{1}{2}\cos \left( \alpha - \beta \right)\]

Now,

\[ - 1 \leq \cos \left( \alpha - \beta \right) \leq 1\]

\[ \Rightarrow - \frac{1}{2} \leq \frac{1}{2}\cos\left( \alpha - \beta \right) \leq \frac{1}{2}\]

\[ \Rightarrow - \frac{1}{2} \leq x \leq \frac{1}{2}\]

\[ \Rightarrow - \frac{1}{2} \leq \cos \alpha \cos \beta \leq \frac{1}{2}\]

\[\text{Hence}, \frac{1}{2}\text{ is the maximum value of }\cos \alpha \cos \beta .\]