Can there exist triangles ABC satisfying the following relations? Write yes or no giving reasons:
$c o s A c o s B = \frac{\sqrt{3}}{4} = s i n A s i n B .$

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Solution

Yes.
$c o s (A - B) = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4}$ = $\frac{\sqrt{3}}{2} ∴ A - B = 30^{0}$
$c o s (A + B)$ = 0 $∴ A + B = 90^{0}$
$∴ A = 60^{0}, B = 30^{0}$
$∴ C = 180^{0} - (A + B) = 90^{0}$

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