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Question Number 108016 by mathdave last updated on 13/Aug/20

Answered by mr W last updated on 13/Aug/20

$$\theta ={\cos }^{-1}x\Rightarrow 0⩽\theta ⩽\pi \Rightarrow 0⩽3\theta ⩽3\pi$$$$\cos \theta =x$$$$\varphi ={\sin }^{-1}2x\Rightarrow -\frac{\pi }{2}⩽\varphi ⩽\frac{\pi }{2}\Rightarrow -\pi ⩽2\varphi ⩽\pi$$$$\sin \varphi =2x=2\cos \theta$$$$2\varphi =3\theta \Rightarrow 0⩽3\theta ⩽\pi \Rightarrow 0⩽\theta ⩽\frac{\pi }{3}\Rightarrow \frac{1}{2}⩽\cos \theta ⩽1$$$$\cos \left(2\varphi \right)=\cos \left(3\theta \right)$$$$1-2{\sin }^2\varphi =4{\cos }^3\theta -3\cos \theta$$$$1-8{\cos }^2\theta =4{\cos }^3\theta -3\cos \theta$$$$\Rightarrow 4{\cos }^3\theta +8{\cos }^2\theta -3\cos \theta -1=0$$$$\Rightarrow (2\cos \theta -1)(2{\cos }^2\theta +5\cos \theta +1)=0$$$$\Rightarrow \cos \theta =\frac{1}{2}$$$$\Rightarrow \cos \theta =\frac{-5\pm \sqrt{17}}{2}$$$$since\frac{1}{2}⩽\cos \theta ⩽1$$$$\Rightarrow x=\cos \theta =\frac{1}{2}\Rightarrow onlyonesolution$$

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