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Question Number 86346 by Power last updated on 28/Mar/20

Commented by MJS last updated on 28/Mar/20

$$-4$$

Commented by jagoll last updated on 28/Mar/20

$$\mathrm{yes}.....\mathrm{i}\mathrm{got}\mathrm{the}\mathrm{same}\mathrm{result}\mathrm{sir}$$

Answered by TANMAY PANACEA. last updated on 28/Mar/20

$$7\theta =2\pi$$$$4\theta =2\pi -3\theta$$$$cos4\theta =cos(2\pi -3\theta )$$$$2{cos}^{2}2\theta -1=4{cos}^3\theta -3cos\theta$$$$2{(2{cos}^2\theta -1)}^2-1=4{cos}^3\theta -3cos\theta$$$$2{(2x^2-1)}^2-1=4x^3-3x$$$$2(4x^4-4x^2+1)-4x^3+3x-1=0$$$$8x^4-4x^3-8x^2+3x+1=0$$$$8x^4-8x^3+4x^3-4x^2-4x^2+4x-x+1$$$$8x^3(x-1)+4x^2(x-1)-4x(x-1)-1(x-1)=0$$$$(x-1)(8x^3+4x^2-4x-1)=0$$$$nowlook8x^3+4x^2-4x-1=0$$$$hasthreerootandonerootiscos\theta$$$$\mathit{c}\mathit{o}\mathit{s}\left(\frac{2\pi }{7}\right)anothercos\left(\frac{4\pi }{7}\right)andcos\left(\frac{6\pi }{7}\right)$$$$cos\left(\frac{2\pi }{7}\right)+cos\left(\frac{4\pi }{7}\right)+cos\left(\frac{6\pi }{7}\right)=\frac{-4}{8}=\frac{-1}{2}$$$$\mathrm{\Sigma }cos\left(\frac{2\pi }{7}\right)cos\left(\frac{4\pi }{7}\right)=\frac{-4}{8}=\frac{-1}{2}$$$$cos\left(\frac{2\pi }{7}\right)cos\left(\frac{4\pi }{7}\right)cos\left(\frac{6\pi }{7}\right)=\frac{1}{8}$$$$\frac{1}{cos\left(\frac{2\pi }{7}\right)}+\frac{1}{cos\left(\frac{4\pi }{7}\right)}+\frac{1}{cos\left(\frac{8\pi }{7}\right)}$$$$look...$$$$cos\left(\frac{8\pi }{7}\right)=cos(\pi +\frac{\pi }{7})=-cos\left(\frac{\pi }{7}\right)$$$$cos\left(\frac{6\pi }{7}\right)=cos(\pi -\frac{\pi }{7})=-cos\left(\frac{\pi }{7}\right)$$$$socos\left(\frac{6\pi }{7}\right)=cos\left(\frac{8\pi }{7}\right)$$$$\frac{1}{cos\left(\frac{2\pi }{7}\right)}+\frac{1}{cos\left(\frac{4\pi }{7}\right)}+\frac{1}{cos\left(\frac{8\pi }{7}\right)}$$$$=\frac{1}{cos\left(\frac{2\pi }{7}\right)}+\frac{1}{cos\left(\frac{4\pi }{7}\right)}+\frac{1}{cos\left(\frac{6\pi }{7}\right)}$$$$=\frac{\mathrm{\Sigma }cos\left(\frac{2\pi }{7}\right)cos\left(\frac{4\pi }{7}\right)}{cos\left(\frac{2\pi }{7}\right)cos\left(\frac{4\pi }{7}\right)cos\left(\frac{6\pi }{7}\right)}$$$$=\frac{\frac{-1}{2}}{\frac{1}{8}}=-4$$$$$$

Commented by Power last updated on 28/Mar/20

$$\mathrm{step}\mathrm{by}\mathrm{step}\mathrm{solution}\mathrm{sir}$$

Commented by jagoll last updated on 28/Mar/20

$$\mathrm{vieta}\mathrm{rule}$$$$\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=-\frac{\mathrm{c}}{\mathrm{d}}=-\left(\frac{-4}{-1}\right)=-4$$$$\mathrm{it}\mathrm{the}\mathrm{answer}$$

Commented by MJS last updated on 28/Mar/20

$$\mathrm{any}\mathrm{steps}\mathrm{by}\mathrm{yourself}\mathrm{Sir}\mathrm{Power}?$$

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