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Question
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Solution
\[\text{ Let }x = \cos \alpha \cos \beta\]
\[ \Rightarrow x = \frac{1}{2}\left[ 2\cos \alpha \cos \beta \right]\]
\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha + \beta \right) + \cos \left( \alpha - \beta \right) \right]\]
\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha - \beta \right) + \cos 90^\circ \right]\]
\[ \Rightarrow x = \frac{1}{2}\cos \left( \alpha - \beta \right)\]
Now,
\[ - 1 \leq \cos \left( \alpha - \beta \right) \leq 1\]
\[ \Rightarrow - \frac{1}{2} \leq \frac{1}{2}\cos\left( \alpha - \beta \right) \leq \frac{1}{2}\]
\[ \Rightarrow - \frac{1}{2} \leq x \leq \frac{1}{2}\]
\[ \Rightarrow - \frac{1}{2} \leq \cos \alpha \cos \beta \leq \frac{1}{2}\]
\[\text{Hence}, \frac{1}{2}\text{ is the maximum value of }\cos \alpha \cos \beta .\]
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