Question

How do you simplify $\cos \left({\sin }^{-1}x\right)$ ?

Answer 1

Step $1$ : Simplify ${\sin }^{-1}x$.

Let us assume, ${\sin }^{-1}\left(x\right)=\theta$

$$\begin{gathered} \Rightarrow \sin \theta =x \\ \Rightarrow {\sin }^2\theta =x^2 \\ \Rightarrow 1-{\cos }^2\theta =x^2 \\ \Rightarrow {\cos }^2\theta =1-x^2 \\ \Rightarrow \cos \theta =\pm \sqrt{1-x^2} \\ \Rightarrow \theta ={\cos }^{-1}\left(\pm \sqrt{1-x^2}\right) \end{gathered}$$ $...\left[si{n}^{2}x+co{s}^{2}x=1\right]$

Step $2$ : Substitute the value of ${\sin }^{-1}x$.

$$\begin{gathered} \cos \left({\sin }^{-1}x\right)=\cos \left(\theta \right) \\ \cos \left({\sin }^{-1}x\right)=\cos \left\{{\cos }^{-1}\right(\pm \sqrt{1-x^2}\left)\right\} \\ \cos \left({\sin }^{-1}x\right)=\pm \sqrt{1-x^2} \end{gathered}$$

Hence, $\cos \left({\sin }^{-1}x\right)$ can be simplified as $\pm \sqrt{1-x^2}$.