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Question Number 175544 by cortano1 last updated on 02/Sep/22

$$\tan {}^6\left(10°\right)+\tan {}^6\left(50°\right)+\tan {}^6\left(70°\right)=?$$

Commented by Frix last updated on 29/Jan/23

$$433$$

Answered by a.lgnaoui last updated on 03/Sep/22

$$posonst=\tan \left(5°\right)\tan {10}^{°}=\tan \left(2t\right){\tan 50}^{°}=\tan ({45}^{°}+t){\tan 70}^{°}=\tan ({45}^{°}+{25}^{°})t^{\prime }=\tan \left(25°\right)$$$${\left(\frac{2t}{1-t^2}\right)}^6+{\left(\frac{1+t}{1-t}\right)}^6+{\left(\frac{1-t^{\prime }}{1+t^{\prime }}\right)}^6$$$$={\left(\frac{1}{1-t}\right)}^6[2^6{t}^6{\left(\frac{1}{1+t}\right)}^6+{(1+t)}^6]+{\left(\frac{1-t^{\prime }}{1+t^{\prime }}\right)}^6$$$$=\frac{1}{{(1-t)}^6}[\frac{64t^6}{{(1+t)}^6}+{(1+t)}^6]$$$$t=0,087t^{{}^{\prime }}=\tan \left(25°\right)=\sqrt{2}-1=0,414$$$$$$$$=\frac{1}{{(0,913)}^6}[\frac{64\times {(0,087)}^6}{{(1,087)}^6}+{(1,087)}^6]+\frac{{(0,586)}^6}{8}=2,847+0,05=2,897$$$$Resultat:$$$$\tan 10+\tan 50+\tan 70=2,897$$$$$$

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