3)
$A=\frac{1}{sin20^{\circ }cos40^{\circ }} +\frac{1}{\sqrt{3} cos20^{\circ }cos40^{\circ }}$
$A=\frac{2cos20^{\circ }}{2cos20^{\circ }sin20^{\circ }cos40^{\circ }}+\frac{2sin20^{\circ }}{(\sqrt{3})2 sin20^{\circ }cos20^{\circ }cos40^{\circ }}$
$A=\frac{2cos20^{\circ }}{sin40^{\circ }cos40^{\circ }}+\frac{2sin20^{\circ }}{(\sqrt{3}) sin40^{\circ }cos40^{\circ }}$
$A=\frac{4cos20^{\circ }}{sin80^{\circ }}+\frac{4sin20^{\circ }}{\sqrt{3} sin80^{\circ }}$
$A=\frac{4\sqrt{3}cos20^{\circ }+4sin20^{\circ } }{\sqrt{3} sin80^{\circ }} $
$A=\frac{8(\frac{\sqrt{3} }{2}cos20^{\circ }+\frac{1}{2}sin20^{\circ }) }{\sqrt{3} sin80^{\circ }} $
$A=\frac{8(cos30^{\circ }cos20^{\circ }+sin30^{\circ }sin20^{\circ })}{\sqrt{3} sin80^{\circ }} $
$A=\frac{8cos10^{\circ }}{\sqrt{3} sin80^{\circ }} $
$A=\frac{8cos10^{\circ }}{\sqrt{3} cos10^{\circ }} $
$A=\frac{8}{\sqrt{3} } $
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$B=cot40^{\circ }+cot230^{\circ }-tan185^{\circ }(tan230^{\circ }-cot50^{\circ })$
$B=cot40^{\circ }+tan40^{\circ }-tan5^{\circ }(cot40^{\circ }-cot50^{\circ })$
$B=\frac{cos40^{\circ }}{sin40^{\circ }} +\frac{sin40^{\circ }}{cos40^{\circ }}-tan5^{\circ }(\frac{cos40^{\circ }}{sin40^{\circ }} -\frac{cos50^{\circ }}{sin50^{\circ }})$
$B=\frac{cos^{2}40^{\circ }+sin^{2}40^{\circ }}{sin40^{\circ }cos40^{\circ }} -tan5^{\circ }(\frac{sin50^{\circ }cos40^{\circ }-cos50^{\circ }sin40^{\circ }}{sin40^{\circ }sin50^{\circ }}) $
$B=\frac{2}{sin80^{\circ }}-tan5^{\circ }(\frac{sin10^{\circ }}{sin40^{\circ }cos40^{\circ }} )$
$B=\frac{2}{sin80^{\circ }}-tan5^{\circ }(\frac{2sin10^{\circ }}{sin80^{\circ }}) $
$B=\frac{2}{cos10^{\circ }}-\frac{sin5^{\circ }}{cos5^{\circ }}(\frac{2sin10^{\circ }}{cos10^{\circ }}) $
$B=\frac{2}{cos10^{\circ }}\left(1-sin10^{\circ }\frac{sin5^{\circ }}{cos5^{\circ }} \right) $
$B=\frac{2}{cos10^{\circ }}\left(1-2sin5^{\circ }cos5^{\circ }\frac{sin5^{\circ }}{cos5^{\circ }} \right) $
$B=\frac{2}{cos10^{\circ }}\left(1-2sin^{2}5^{\circ }\right) $
$B=\frac{2}{cos10^{\circ }}\left(cos10^{\circ }\right) $
$B=2$
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$AB=(\frac{8}{\sqrt{3} } )(2)$
$AB=\frac{16}{\sqrt{3} } $
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