Question

Write in simplest form ${\sin }^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right]$

Answer 1

$\begin{array}{c}{\sin }^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right]\\ letx=\left[{\cos }^2\theta \right]\\ \Rightarrow {\sin }^{-1}\left(\frac{\sqrt{1+{\cos }^2\theta }+\sqrt{{\cos }^2\theta }}{2}\right)\\ \Rightarrow {\sin }^{-1}\left[\frac{\cos \theta }{\sqrt{2}}+\frac{\sin \theta }{\sqrt{2}}\right]\\ \Rightarrow {\sin }^{-1}\left[\cos \theta \cdot \sin {45}^0+\sin \theta \cdot \cos {45}^0\right]\\ \Rightarrow {\sin }^{-1}\left[\sin \left(\theta +{45}^0\right)\right]\\ \therefore \theta +{45}^0\\ \Rightarrow \frac{{\cos }^{-1}x}{2}+{45}^0\end{array}$