Question

Write sin^-1( x+ square root 1-x²/ root 2 in the simplest form

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{y}={\sin }^{-1}\left[\frac{\mathrm{x}+\sqrt{1-{\mathrm{x}}^2}}{\sqrt{2}}\right] \\ \Rightarrow \mathrm{y}={\sin }^{-1}\left[\frac{1}{\sqrt{2}}\times \mathrm{x}+\frac{1}{\sqrt{2}}\times \sqrt{1-{\mathrm{x}}^2}\right] \\ \Rightarrow \mathrm{y}={\sin }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{4}\right)\times \mathrm{x}+\cos \left(\frac{\mathrm{\pi }}{4}\right)\times \sqrt{1-{\mathrm{x}}^2}\right] \\ \mathrm{Let}\mathrm{x}=\cos \mathrm{\alpha } \\ \mathrm{Then}\sin \mathrm{\alpha }=\sqrt{1-{\cos }^2\mathrm{\alpha }}=\sqrt{1-{\mathrm{x}}^2} \\ \mathrm{Now},\mathrm{y}={\sin }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{4}\right)\times \cos \mathrm{\alpha }+\cos \left(\frac{\mathrm{\pi }}{4}\right)\times \mathrm{sin\alpha }\right] \\ \Rightarrow \mathrm{y}={\sin }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{4}+\mathrm{\alpha }\right)\right] \\ \Rightarrow \mathrm{y}=\frac{\mathrm{\pi }}{4}+\mathrm{\alpha } \\ \Rightarrow \boxed{\mathbf{y}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{x}} \end{gathered}$$