Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 18742 by Tinkutara last updated on 31/Jul/17

The value of cot16°cot44° + cot44°cot76° − cot76°cot16° is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cot16}°\mathrm{cot44}°\:+\:\mathrm{cot44}°\mathrm{cot76}° \\ $$$$−\:\mathrm{cot76}°\mathrm{cot16}°\:\mathrm{is} \\ $$

Answered by Tinkutara last updated on 31/Jul/17

((3 + cot 76° cot 16°)/(cot 76° + cot 16°)) = ((3 sin 76° sin 16° + cos 76° cos 16°)/(sin (76° + 16°))) = ((2 sin 76° sin 16° + cos (76° − 16°))/(sin 92°)) = ((cos 60° − cos 92° + cos 60°)/(sin 92°)) = ((1 − cos 92°)/(sin 92°)) = tan 46° = cot 44° Now, ((3 + cot 76° cot 16°)/(cot 76° + cot 16°)) = cot 44° cot16°cot44° + cot44°cot76° − cot76°cot16° = 3

$$\frac{\mathrm{3}\:+\:\mathrm{cot}\:\mathrm{76}°\:\mathrm{cot}\:\mathrm{16}°}{\mathrm{cot}\:\mathrm{76}°\:+\:\mathrm{cot}\:\mathrm{16}°} \\ $$$$=\:\frac{\mathrm{3}\:\mathrm{sin}\:\mathrm{76}°\:\mathrm{sin}\:\mathrm{16}°\:+\:\mathrm{cos}\:\mathrm{76}°\:\mathrm{cos}\:\mathrm{16}°}{\mathrm{sin}\:\left(\mathrm{76}°\:+\:\mathrm{16}°\right)} \\ $$$$=\:\frac{\mathrm{2}\:\mathrm{sin}\:\mathrm{76}°\:\mathrm{sin}\:\mathrm{16}°\:+\:\mathrm{cos}\:\left(\mathrm{76}°\:−\:\mathrm{16}°\right)}{\mathrm{sin}\:\mathrm{92}°} \\ $$$$=\:\frac{\mathrm{cos}\:\mathrm{60}°\:−\:\mathrm{cos}\:\mathrm{92}°\:+\:\mathrm{cos}\:\mathrm{60}°}{\mathrm{sin}\:\mathrm{92}°} \\ $$$$=\:\frac{\mathrm{1}\:−\:\mathrm{cos}\:\mathrm{92}°}{\mathrm{sin}\:\mathrm{92}°}\:=\:\mathrm{tan}\:\mathrm{46}°\:=\:\mathrm{cot}\:\mathrm{44}° \\ $$$$\mathrm{Now},\:\frac{\mathrm{3}\:+\:\mathrm{cot}\:\mathrm{76}°\:\mathrm{cot}\:\mathrm{16}°}{\mathrm{cot}\:\mathrm{76}°\:+\:\mathrm{cot}\:\mathrm{16}°}\:=\:\mathrm{cot}\:\mathrm{44}° \\ $$$$\mathrm{cot16}°\mathrm{cot44}°\:+\:\mathrm{cot44}°\mathrm{cot76}°\:−\:\mathrm{cot76}°\mathrm{cot16}°\:=\:\mathrm{3} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com