Question Number 147651 by iloveisrael last updated on 22/Jul/21
$$\:\mathrm{tan}\:\mathrm{1}°+\mathrm{tan}\:\mathrm{5}°+\mathrm{tan}\:\mathrm{9}°+…+\mathrm{tan}\:\mathrm{173}°+\mathrm{tan}\:\mathrm{177}°=? \\ $$
Commented by mr W last updated on 22/Jul/21
$$=\mathrm{45} \\ $$
Commented by iloveisrael last updated on 23/Jul/21
$$\mathrm{is}\:\mathrm{the}\:\mathrm{formula}\: \\ $$$$\:\mathrm{tan}\:\alpha+\mathrm{tan}\:\left(\mathrm{60}°−\alpha\right)+\mathrm{tan}\:\left(\mathrm{60}°+\alpha\right)=\mathrm{3tan}\:\left(\mathrm{3}\alpha\right) \\ $$$$\:\mathrm{correct}\:? \\ $$
Answered by mathmax by abdo last updated on 22/Jul/21
$$\mathrm{S}=\mathrm{tan}\left(\frac{\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{5}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+….+\mathrm{tan}\left(\frac{\mathrm{169}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{173}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{177}\pi}{\mathrm{180}}\right) \\ $$$$\mathrm{S}=\mathrm{tan}\left(\frac{\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{5}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+…+\mathrm{tan}\left(\pi−\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\pi−\frac{\mathrm{5}\pi}{\mathrm{180}}\right) \\ $$$$+\mathrm{tan}\left(\pi−\frac{\pi}{\mathrm{190}}\right)\:=\mathrm{0} \\ $$