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Calculate-3-cosec-20-sec-20-




Question Number 140196 by mathdanisur last updated on 05/May/21
Calculate: (√3) cosec 20°−sec 20°
$${Calculate}:\:\sqrt{\mathrm{3}}\:{cosec}\:\mathrm{20}°−{sec}\:\mathrm{20}° \\ $$
Answered by liberty last updated on 05/May/21
 ((√3)/(sin 20°))−(1/(cos 20°)) =   ((2((√3) cos 20°−sin 20°))/(sin 40°)) =  ((4(((√3)/2) cos 20°−(1/2)sin 20°))/(sin 40°)) =  ((4(cos 30° cos 20°−sin 30° sin 20°))/(sin 40°)) =  ((4cos 50°)/(sin 40°)) = 4
$$\:\frac{\sqrt{\mathrm{3}}}{\mathrm{sin}\:\mathrm{20}°}−\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{20}°}\:=\: \\ $$$$\frac{\mathrm{2}\left(\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{20}°−\mathrm{sin}\:\mathrm{20}°\right)}{\mathrm{sin}\:\mathrm{40}°}\:= \\ $$$$\frac{\mathrm{4}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{cos}\:\mathrm{20}°−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{20}°\right)}{\mathrm{sin}\:\mathrm{40}°}\:= \\ $$$$\frac{\mathrm{4}\left(\mathrm{cos}\:\mathrm{30}°\:\mathrm{cos}\:\mathrm{20}°−\mathrm{sin}\:\mathrm{30}°\:\mathrm{sin}\:\mathrm{20}°\right)}{\mathrm{sin}\:\mathrm{40}°}\:= \\ $$$$\frac{\mathrm{4cos}\:\mathrm{50}°}{\mathrm{sin}\:\mathrm{40}°}\:=\:\mathrm{4} \\ $$

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