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Question Number 46473 by peter frank last updated on 27/Oct/18

$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\theta$$$$\mathrm{which}\mathrm{satisfy}\mathrm{the}$$$$\mathrm{equation}$$$$\cos \theta \mathrm{x}+\cos (\mathrm{x}+2)\theta =\cos \theta$$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Oct/18

$$2cos\left(\frac{x\theta +x\theta +2\theta }{2}\right)cos\left(\frac{x\theta +2\theta -x\theta }{2}\right)=cos\theta$$$$2cos(x\theta +\theta )cos\theta -cos\theta =0$$$$cos\theta \{2cos(x\theta +\theta )-1\}=0$$$$eithercos\theta =0cos\theta =cos\frac{\pi }{2}$$$$so\theta =2n\pi \pm \frac{\pi }{2}$$$$orcos(x\theta +\theta )=\frac{1}{2}=cos\frac{\pi }{3}$$$$\theta (x+1)=2n\pi \pm \frac{\pi }{3}$$$$\theta =\frac{1}{x+1}[2n\pi \pm \frac{\pi }{3}]$$

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