しばらく下書きのまま放置してたけどとりあえず投稿してみる2つ目と3つ目の左辺はMathematicaでやったのをそのままコピペした
一応追記:Log[x]でlogx(自然対数)Cos[x]やSin[x]でcosx, sinxSqrt[x]で√xの意味[ ]はガウス記号じゃないですπは円周率

-1/(2log((1+√5)/2))*(log(sin(π/5))-log(sin(2π/5))-log(sin(3π/5))+log(sin(4π/5)))=1

(1/Log[3 + Sqrt[10]])(Log[Cos[π/40]] + Log[Cos[(3 π)/40]] -
  Log[Cos[(7 π)/40]] + Log[Cos[(9 π)/40]] - Log[Sin[π/40]] -

  Log[Sin[(3 π)/40]] + Log[Sin[(7 π)/40]] - Log[Sin[(9 π)/40]])=2



(1/Log[9 + Sqrt[82]])(Log[Cos[π/328]] + Log[Cos[(3 π)/328]] -
  Log[Cos[(5 π)/328]] - Log[Cos[(7 π)/328]] + Log[Cos[(9 π)/328]] +
  Log[Cos[(11 π)/328]] + Log[Cos[(13 π)/328]] - Log[Cos[(15 π)/328]] -
  Log[Cos[(17 π)/328]] + Log[Cos[(19 π)/328]] - Log[Cos[(21 π)/328]] +
  Log[Cos[(23 π)/328]] + Log[Cos[(25 π)/328]] + Log[Cos[(27 π)/328]] +
  Log[Cos[(29 π)/328]] + Log[Cos[(31 π)/328]] + Log[Cos[(33 π)/328]] +
  Log[Cos[(35 π)/328]] - Log[Cos[(37 π)/328]] + Log[Cos[(39 π)/328]] -
  Log[Cos[(43 π)/328]] - Log[Cos[(45 π)/328]] - Log[Cos[(47 π)/328]] +
  Log[Cos[(49 π)/328]] - Log[Cos[(51 π)/328]] + Log[Cos[(53 π)/328]] -
  Log[Cos[(55 π)/328]] + Log[Cos[(57 π)/328]] - Log[Cos[(59 π)/328]] -
  Log[Cos[(61 π)/328]] - Log[Cos[(63 π)/328]] - Log[Cos[(65 π)/328]] +
  Log[Cos[(67 π)/328]] + Log[Cos[(69 π)/328]] - Log[Cos[(71 π)/328]] +
  Log[Cos[(73 π)/328]] + Log[Cos[(75 π)/328]] - Log[Cos[(77 π)/328]] -
  Log[Cos[(79 π)/328]] + Log[Cos[(81 π)/328]] - Log[Sin[π/328]] -
  Log[Sin[(3 π)/328]] + Log[Sin[(5 π)/328]] + Log[Sin[(7 π)/328]] -
  Log[Sin[(9 π)/328]] - Log[Sin[(11 π)/328]] - Log[Sin[(13 π)/328]] +
  Log[Sin[(15 π)/328]] + Log[Sin[(17 π)/328]] - Log[Sin[(19 π)/328]] +
  Log[Sin[(21 π)/328]] - Log[Sin[(23 π)/328]] - Log[Sin[(25 π)/328]] -
  Log[Sin[(27 π)/328]] - Log[Sin[(29 π)/328]] - Log[Sin[(31 π)/328]] -
  Log[Sin[(33 π)/328]] - Log[Sin[(35 π)/328]] + Log[Sin[(37 π)/328]] -
  Log[Sin[(39 π)/328]] + Log[Sin[(43 π)/328]] + Log[Sin[(45 π)/328]] +
  Log[Sin[(47 π)/328]] - Log[Sin[(49 π)/328]] + Log[Sin[(51 π)/328]] -
  Log[Sin[(53 π)/328]] + Log[Sin[(55 π)/328]] - Log[Sin[(57 π)/328]] +
  Log[Sin[(59 π)/328]] + Log[Sin[(61 π)/328]] + Log[Sin[(63 π)/328]] +
  Log[Sin[(65 π)/328]] - Log[Sin[(67 π)/328]] - Log[Sin[(69 π)/328]] +
  Log[Sin[(71 π)/328]] - Log[Sin[(73 π)/328]] - Log[Sin[(75 π)/328]] +

  Log[Sin[(77 π)/328]] + Log[Sin[(79 π)/328]] - Log[Sin[(81 π)/328]])=4