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Question Number 15641 by tawa tawa last updated on 12/Jun/17

Show that. (√3)(cosec20) − sec20 = 4

$$\mathrm{Show}\:\mathrm{that}. \\ $$$$\sqrt{\mathrm{3}}\left(\mathrm{cosec20}\right)\:−\:\mathrm{sec20}\:=\:\mathrm{4} \\ $$

Answered by mrW1 last updated on 12/Jun/17

((√3)/(sin 20))−(1/(cos 20))=(((√3)cos 20−1sin 20)/(sin 20cos 20)) =((4(((√3)/2)cos 20−(1/2)sin 20))/(sin 40)) =((4(cos 30cos 20−sin 30sin 20))/(sin 40)) =((4cos 50)/(sin 40)) =((4sin 40)/(sin 40)) =4

$$\frac{\sqrt{\mathrm{3}}}{\mathrm{sin}\:\mathrm{20}}−\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{20}}=\frac{\sqrt{\mathrm{3}}\mathrm{cos}\:\mathrm{20}−\mathrm{1sin}\:\mathrm{20}}{\mathrm{sin}\:\mathrm{20cos}\:\mathrm{20}} \\ $$$$=\frac{\mathrm{4}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{cos}\:\mathrm{20}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{20}\right)}{\mathrm{sin}\:\mathrm{40}} \\ $$$$=\frac{\mathrm{4}\left(\mathrm{cos}\:\mathrm{30cos}\:\mathrm{20}−\mathrm{sin}\:\mathrm{30sin}\:\mathrm{20}\right)}{\mathrm{sin}\:\mathrm{40}} \\ $$$$=\frac{\mathrm{4cos}\:\mathrm{50}}{\mathrm{sin}\:\mathrm{40}} \\ $$$$=\frac{\mathrm{4sin}\:\mathrm{40}}{\mathrm{sin}\:\mathrm{40}} \\ $$$$=\mathrm{4} \\ $$

Commented by tawa tawa last updated on 12/Jun/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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