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My sister asked me such a trigonometric identity (her high school challenging problem):

prove:

$$\frac1{\sin40 ^\circ}+\tan10^\circ=\sqrt{3}.$$

I found that this is really true (surprising... with a calculator), but as an undergraduate equipped with calculus and linear algebra, I have no idea how to attack this problem. Thanks in advance for any help!

Lion Heart
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youknowwho
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3 Answers3

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We need to prove that: $$\frac{1}{\sin40^{\circ}}=\tan60^{\circ}-\tan10^{\circ}$$ or $$\frac{1}{\sin40^{\circ}}=\frac{\sin50^{\circ}}{\cos60^{\circ}\cos10^{\circ}}$$ 0r $$\cos10^{\circ}=2\sin40^{\circ}\cos40^{\circ}$$ or $$\cos10^{\circ}=\sin80^{\circ}.$$

Michael Rozenberg
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$$\frac1{\sin40 ^\circ}+\tan10^\circ=\frac{2\cos40^\circ}{2\times\sin40 ^\circ\times \cos40^\circ=\sin80^\circ=\cos10^\circ}+\frac {\sin10^\circ=\cos80^\circ}{\cos10^\circ}= \frac{2\times\cos40^\circ + \cos80^\circ}{\cos10^\circ}=\frac{\cos40^\circ + \cos40^\circ + \cos80^\circ}{\cos10^\circ}=\frac{\cos40^\circ+2\times\cos60^\circ\times\cos20^\circ }{\cos10^\circ}=\frac{\cos40^\circ+\cos20^\circ}{\cos10^\circ}=\frac{2\times\cos30^\circ \times\cos10^\circ}{\cos10^\circ} = \sqrt{3}.$$

Lion Heart
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Replace $\sqrt{3}$ with $2 \sin{60^\circ}$. We want to show $$\frac1{\sin40 ^\circ}+\tan10^\circ=2 \sin{60^\circ}.$$

Let $q = e^{2 i \pi / 360}$ (A nome of one degree) then ​$$\sin(\theta^\circ) = \frac{q^\theta - q^{-\theta}}{2i}$$ and ​$$\tan(\theta^\circ) = - i \frac{q^{\theta} - q^{-\theta}}{q^{\theta} + q^{-\theta}}$$ so we can rewrite our equation in terms of the nome:

$$\frac{2i}{q^{40} - q^{-40}} - i \frac{q^{10} - q^{-10}}{q^{10} + q^{-10}} - 2 \frac{q^{60} - q^{-60}}{2i} = 0$$

multiplying this out and then reducing it modulo the cyclotomic polynomial $\Phi_{360}(q) = q^{96} + q^{84} - q^{60} - q^{48} - q^{36} + q^{12} + 1$ gives zero:

? r = (2*I)/(q^40 - q^(-40)) - I*(q^10-q^(-10))/(q^10+q^(-10)) - 2*(q^60-q^(-60))/(2*I)
% = (-q^200 + q^140 - q^120 - q^80 + q^60 - 1)/(I*q^140 - I*q^60)
? Mod(r, q^96 + q^84 - q^60 - q^48 - q^36 + q^12 + 1)
% = Mod(0, q^96 + q^84 - q^60 - q^48 - q^36 + q^12 + 1)

this proves the identity.