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Question Number 147651 by iloveisrael last updated on 22/Jul/21

tan 1°+tan 5°+tan 9°+...+tan 173°+tan 177°=?

$$\:\mathrm{tan}\:\mathrm{1}°+\mathrm{tan}\:\mathrm{5}°+\mathrm{tan}\:\mathrm{9}°+...+\mathrm{tan}\:\mathrm{173}°+\mathrm{tan}\:\mathrm{177}°=? \\ $$

Commented by mr W last updated on 22/Jul/21

=45

$$=\mathrm{45} \\ $$

Commented by iloveisrael last updated on 23/Jul/21

is the formula tan α+tan (60°−α)+tan (60°+α)=3tan (3α) correct ?

$$\mathrm{is}\:\mathrm{the}\:\mathrm{formula}\: \\ $$$$\:\mathrm{tan}\:\alpha+\mathrm{tan}\:\left(\mathrm{60}°−\alpha\right)+\mathrm{tan}\:\left(\mathrm{60}°+\alpha\right)=\mathrm{3tan}\:\left(\mathrm{3}\alpha\right) \\ $$$$\:\mathrm{correct}\:? \\ $$

Answered by mathmax by abdo last updated on 22/Jul/21

S=tan((π/(180)))+tan(((5π)/(180)))+tan(((9π)/(180)))+....+tan(((169π)/(180)))+tan(((173π)/(180)))+tan(((177π)/(180))) S=tan((π/(180)))+tan(((5π)/(180)))+tan(((9π)/(180)))+...+tan(π−((9π)/(180)))+tan(π−((5π)/(180))) +tan(π−(π/(190))) =0

$$\mathrm{S}=\mathrm{tan}\left(\frac{\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{5}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+....+\mathrm{tan}\left(\frac{\mathrm{169}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{173}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{177}\pi}{\mathrm{180}}\right) \\ $$$$\mathrm{S}=\mathrm{tan}\left(\frac{\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{5}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+...+\mathrm{tan}\left(\pi−\frac{\mathrm{9}\pi}{\mathrm{180}}\right)+\mathrm{tan}\left(\pi−\frac{\mathrm{5}\pi}{\mathrm{180}}\right) \\ $$$$+\mathrm{tan}\left(\pi−\frac{\pi}{\mathrm{190}}\right)\:=\mathrm{0} \\ $$

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