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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

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

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