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Find-1-sin-10-4-sin-70-




Question Number 185668 by Shrinava last updated on 25/Jan/23
Find:     (1/(sin 10°)) − 4 sin 70° = ?
$$\mathrm{Find}:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{10}°}\:−\:\mathrm{4}\:\mathrm{sin}\:\mathrm{70}°\:=\:? \\ $$
Answered by Ezzat last updated on 25/Jan/23
=((1−4sin 10° sin70° )/(sin 10°))   =((1−4×(1/2)(cos(10°−70°)−cos(10°+70°)))/(sin 10°))  =((1−2(cos(60°)−cos80°))/(sin 10°))  =((1−2((1/2)−cos80°))/(sin10°))  =((1−1+2cos80°)/(sin10°))  = 2   ((cos80°)/(cos80°))   =  2
$$=\frac{\mathrm{1}−\mathrm{4}{sin}\:\mathrm{10}°\:{sin}\mathrm{70}°\:}{{sin}\:\mathrm{10}°}\: \\ $$$$=\frac{\mathrm{1}−\mathrm{4}×\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\left(\mathrm{10}°−\mathrm{70}°\right)−{cos}\left(\mathrm{10}°+\mathrm{70}°\right)\right)}{{sin}\:\mathrm{10}°} \\ $$$$=\frac{\mathrm{1}−\mathrm{2}\left({cos}\left(\mathrm{60}°\right)−{cos}\mathrm{80}°\right)}{{sin}\:\mathrm{10}°} \\ $$$$=\frac{\mathrm{1}−\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}−{cos}\mathrm{80}°\right)}{{sin}\mathrm{10}°} \\ $$$$=\frac{\mathrm{1}−\mathrm{1}+\mathrm{2}{cos}\mathrm{80}°}{{sin}\mathrm{10}°} \\ $$$$=\:\mathrm{2}\:\:\:\frac{{cos}\mathrm{80}°}{{cos}\mathrm{80}°}\:\:\:=\:\:\mathrm{2} \\ $$
Answered by greougoury555 last updated on 25/Jan/23
   ((1−4sin 70° sin 10°)/(sin 10°))    =((1+2(cos 80°−cos 60°))/(sin 10°))  = 2
$$\:\:\:\frac{\mathrm{1}−\mathrm{4sin}\:\mathrm{70}°\:\mathrm{sin}\:\mathrm{10}°}{\mathrm{sin}\:\mathrm{10}°} \\ $$$$\:\:=\frac{\mathrm{1}+\mathrm{2}\left(\mathrm{cos}\:\mathrm{80}°−\mathrm{cos}\:\mathrm{60}°\right)}{\mathrm{sin}\:\mathrm{10}°} \\ $$$$=\:\mathrm{2} \\ $$

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