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Question Number 193676 by lapache last updated on 18/Jun/23

$$Solve$$$$arcsin\left(2x\right)+arcsin\left(x\sqrt{3}\right)=arcsin\left(x\right)$$

Answered by Frix last updated on 18/Jun/23

$${\sin }^{-1}t\in \mathbb{R}\iff -1⩽t⩽1$$$$f\left(x\right)={\sin }^{-1}2x+{\sin }^{-1}\sqrt{3}x-{\sin }^{-1}x$$$$-\frac{1}{2}⩽x⩽\frac{1}{2}$$$$f^{\prime }\left(x\right)=\frac{2}{\sqrt{1-4x^2}}+\frac{\sqrt{3}}{\sqrt{1-3x^2}}-\frac{1}{\sqrt{1-x^2}}>0\mathrm{\forall }x\in (-\frac{1}{2};\frac{1}{2})$$$$\Rightarrow \mathrm{only}\mathrm{solution}\mathrm{is}x=0$$

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