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Question Number 43643 by gunawan last updated on 13/Sep/18

$$\mathrm{If}{\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x,$$$$\mathrm{then}x=$$

Answered by behi83417@gmail.com last updated on 13/Sep/18

$$x=\frac{1}{2}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Sep/18

$${sin}^{-1}(1-x)={cos}^{-1}x-{sin}^{-1}x$$$$sin\alpha =x$$$$sin(-\alpha )=-x$$$${sin}^{-1}(-x)=-\alpha$$$$sin\alpha =x$$$$cos(\frac{\mathrm{\Pi }}{2}-\alpha )=x$$$$\frac{\mathrm{\Pi }}{2}-\alpha ={cos}^{-1}x$$$${sin}^{-1}(1-x)=\frac{\mathrm{\Pi }}{2}-\alpha -\alpha$$$$sin(\frac{\mathrm{\Pi }}{2}-2\alpha )=1-x$$$$cos2\alpha =1-x$$$$1-2{sin}^2\alpha =1-x$$$$1-2x^2=1-x$$$$-2x^2+x=0$$$$-x(2x-1)=0$$$$x=0orx=\frac{1}{2}$$$$$$$$$$$$$$

Answered by Joel578 last updated on 13/Sep/18

$$\cos ({\sin }^{-1}x+{\sin }^{-1}(1-x))=\cos \left({\cos }^{-1}x\right)$$$$\cos \left({\sin }^{-1}x\right).\cos \left({\sin }^{-1}(1-x)\right)-\sin \left({\sin }^{-1}x\right).\sin \left({\sin }^{-1}(1-x)\right)=x$$$$\left(\sqrt{1-x^2}\right)\left(\sqrt{2x-x^2}\right)-x(1-x)=x$$$$\sqrt{(1-x^2)(2x-x^2)}=2x-x^2$$$$(1-x^2)(2x-x^2)={(2x-x^2)}^2$$$$2x-x^2-2x^3+x^4=4x^2-4x^3+x^4$$$$2x^3-5x^2+2x=0$$$$x(2x-1)(x-2)=0$$$$x=0\vee x=\frac{1}{2}\vee x=2\to {\mathrm{didn}}^{\prime }\mathrm{t}\mathrm{satisfy}$$$$\therefore x=\{0,\frac{1}{2}\}$$

Commented by gunawan last updated on 13/Sep/18

$$\mathrm{Nice}$$$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}$$

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