I have seen a question asked on yahoo asking to find the value of
$\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees)
I have seen various results concerning products of trigonometric values such as
$(1 + \tan1^{\circ}) (1 + \tan2^{\circ}) … (1 + \tan45^{\circ}) = 2^{23} \\$
$\prod_{k=1}^{89} \sin(k \pi/180) = -\prod_{k=-89}^{89} (\sin(k \pi/180)-1) = 360/2^{180}$
I have tried doing various things with roots of unity but nothing much that hasn't been done before. Does anyone have any ideas?
Thanks
Edit: Including some other statements which I consider equivalent to finding this answer
$\cos 1^{\circ} … \cos 45^{\circ}\\
\sin 1^{\circ} … \sin45^{\circ} \\
(1 + \tan 46^{\circ})…(1+\tan 89^{\circ}) \\
(\sin 1^{\circ} + \cos 1^{\circ})…(\sin 45^{\circ} + \cos45^{\circ}) \\
(\sin 2^{\circ} + 1)(\sin 4^{\circ} + 1)….(\sin 88^{\circ} + 1) \\
(1 + \tan^2 1^{\circ})(1 + \tan^2 2^{\circ}) .. (1 + \tan^2 45^{\circ})$
Best Answer
I considered the more general formula : $$T(m):=\prod_{k=1}^m \tan\left(\frac{k\pi}{4m}\right)$$ and noticed that the result for small values of $m$ was solution of a polynomial of degree $\le m$. For $m=45$ I found that the answer was solution of this irreducible polynomial of degree $24$ : $$1\\- 3256701697315828896312\,x^1 - 325994294876282580655116\,x^2 + 7220097128841103979624568\,x^3 + 112578453555034444841119842\,x^4 + 493898299320136273975435032\,x^5 + 649061666980531722406164708\,x^6 - 840700351973464244018822232\,x^7 -2457988129238279755530778353\,x^8 - 138286882106888055215208624\,x^9 +2474525072938192662606171624\,x^{10}+326024084648343835216068912\,x^{11} - 1088043811994145989051965476\,x^{12} + 5147738954805237173669808\,x^{13} +182273284200850360076819304\,x^{14} - 33045263177263307887100976\,x^{15} + 677463542076505961377071\,x^{16} +170537100491574073221480\,x^{17} - 6714674580553776884700\,x^{18} - 128584156182235814952\,x^{19} - 339010000890501150\,x^{20}\\ +776030507612856\,x^{21} - 397610115660\,x^{22}\\ + 37004040\,x^{23} + 6561\,x^{24}$$
This is an 'experimental' result (no proof) but rather satisfying from the 'not nice' point of view ! ;-)