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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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