MHB Trigonometric of tangent and sine functions

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The discussion focuses on simplifying the expression involving tangent and sine functions: \(\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right). A user named Dan shares a numerical approximation of the result, approximately -8.7083, and asks if they are close to the correct answer. The conversation revolves around verifying this simplification and discussing the properties of the trigonometric functions involved. The overall goal is to confirm the accuracy of the numerical result and the simplification process.
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Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
 
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I got about -8.7083121069873265814843145114959219999438416220762493062309701968483080116471438626714199913273209202440106287205185869613863265722279384849632901398226721432702744787955199077781973792055631357781755415456733331388124469116498805756011983384506433640247288593850820089590928253650738597742412754348439402301589178427587352708579714081417809559865987679591010005488147679340458470850085945675037017144904042287279335497160556553452963874716319026939725908000014025345344112227718546669183167834689374961602264946845122520973969612465747159538537772604016219227651262298967156522926816165972425283143645081705710618441263404832174787619393871431981552587401078586609898568054592224581007986448669288148444215128498186229896741206349315952054866256984612864167199842472866575465952447085845790874114020207148838309575537421068097653460699858226519033408205869208586639796284569971790165922274544500088737294024496093237634962135759638417523292668843921768916664652250085254395250841082981914897137469833874653795905403859395226103590166597307639852476273392879083873723649605334228340639098097814440444549718831560592044450598723461692804007733588937741590648889021935199757480637837660523461803937847078950866107512314217003731198786042996011750451712135884514444878781114357178952544416819768594073353584110967734337432198244086867356343864416552649586451520413569547092694404444532795994994438271445333716459116502792251091594744077024600155160457563503684373664211499889132746530159707148096441069149624420565803877227318131627873671715105932608979648146278844613590157353436714676787933231516958800501704045205913556958442328349254532043183475963879427381686337157339601066369199608533673687386762949721994907024739645071023865884051877078161059364899178505753714909479228031574944638783941268199880771092085126999780242627307210910698666574484989827761671955524479251591513340499083326461726929118742316319408632872884749801756534634165971359295929387392681708974873301182019170001880197471302122860499952496977...

Am I close?

-Dan
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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