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Question Number 141383 by Willson last updated on 18/May/21

$${\sin }^2\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{2}\right)=1-\frac{1}{2}\sqrt{1-{\mathrm{x}}^2}$$$$\mathrm{find}\mathrm{f}\left(\mathrm{x}\right)$$

Answered by mr W last updated on 18/May/21

$${\sin }^2\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{2}\right)=1-\frac{1}{2}\sqrt{1-{\mathrm{x}}^2}$$$$1-{\sin }^2\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{2}\right)=\frac{1}{2}\sqrt{1-{\mathrm{x}}^2}$$$${\cos }^2\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{2}\right)=\frac{1}{2}\sqrt{1-{\mathrm{x}}^2}$$$${2\cos }^2\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{2}\right)-1=\sqrt{1-{\mathrm{x}}^2}-1$$$$\cos \left(f\left(x\right)\right)=\sqrt{1-{\mathrm{x}}^2}-1$$$$\Rightarrow f\left(x\right)={\cos }^{-1}(\sqrt{1-{\mathrm{x}}^2}-1)$$

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