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Question Number 209781 by alusto22 last updated on 21/Jul/24

   ((1+sin 40°−cos 40°)/(1+sin 40°+cos 40°)) =?

$$\:\:\:\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{40}°−\mathrm{cos}\:\mathrm{40}°}{\mathrm{1}+\mathrm{sin}\:\mathrm{40}°+\mathrm{cos}\:\mathrm{40}°}\:=? \\ $$

Answered by efronzo1 last updated on 21/Jul/24

    ((1+sin 40°−(1−2sin^2 20°))/(1+sin 40°+2cos^2 20−1))     = ((2sin 20°(cos 20°+sin 20°))/(2cos 20°(cos 20°+sin 20°)))    = tan 20°

$$\:\:\:\:\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{40}°−\left(\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} \mathrm{20}°\right)}{\mathrm{1}+\mathrm{sin}\:\mathrm{40}°+\mathrm{2cos}\:^{\mathrm{2}} \mathrm{20}−\mathrm{1}}\: \\ $$$$\:\:=\:\frac{\mathrm{2sin}\:\mathrm{20}°\left(\mathrm{cos}\:\mathrm{20}°+\mathrm{sin}\:\mathrm{20}°\right)}{\mathrm{2cos}\:\mathrm{20}°\left(\mathrm{cos}\:\mathrm{20}°+\mathrm{sin}\:\mathrm{20}°\right)} \\ $$$$\:\:=\:\mathrm{tan}\:\mathrm{20}°\: \\ $$

Answered by Frix last updated on 22/Jul/24

((1+sin 2x −cos 2x)/(1+sin 2x +cos 2x))=...  =((2sin x cos x +2sin^2  x)/(2sin x cos x +2cos^2  x))=  =((sin x)/(cos x))=tan x

$$\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}\:−\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}\:+\mathrm{cos}\:\mathrm{2}{x}}=... \\ $$$$=\frac{\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}\:+\mathrm{2sin}^{\mathrm{2}} \:{x}}{\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}\:+\mathrm{2cos}^{\mathrm{2}} \:{x}}= \\ $$$$=\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}=\mathrm{tan}\:{x} \\ $$

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