Question

A+B=π/2,prove that the maximum value of cosA*cosB is 1/2

Answer 1

Hey There!

Here, we will use one simple identity:
$2\sin\theta \cos\theta = \sin 2\theta$

Now, let's come to your question:

$A+B=\frac{\pi}{2} \\ \\ \implies B=\frac{\pi}{2}-A$


Now,

$\cos A \cos B \\ \\ = \cos A \cos \left(\frac{\pi}{2}-A\right) \\ \\ = \cos A \sin A \\ \\ = \frac{1}{2} \times 2\sin A \cos A \\ \\ = \frac{1}{2} \sin 2A$


Now, maximum value of sin function is 1.
That is:

$sin 2A \leqslant 1 \\ \\ \frac{1}{2}\sin 2A \leqslant \frac{1}{2}$

Thus,maximum value is $\frac{1}{2}$


Hope it helps
Purva
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