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Question Number 106214 by john santu last updated on 03/Aug/20

$$\mathrm{solve}:\mathrm{arc}\cos (\mathrm{x}-1)=2\mathrm{arc}\cos \left(\mathrm{x}\right)$$$$\mathrm{where}\mathrm{x}\mathrm{is}\mathrm{real}.$$

Answered by bobhans last updated on 03/Aug/20

$$\Rightarrow \cos \left(\mathrm{arc}\cos (\mathrm{x}-1)\right)=\cos \left(2\mathrm{arc}\cos \left(\mathrm{x}\right)\right)$$$$\Rightarrow \mathrm{x}-1=2\cos {}^2\left(\mathrm{arc}\cos \left(\mathrm{x}\right)\right)-1$$$$\Rightarrow \mathrm{x}-1={2\mathrm{x}}^2-1$$$$\to \{\begin{array}{l}\mathrm{x}=0\mathrm{or}\\ \mathrm{x}=\frac{1}{2}\end{array}$$

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