Question

How to transform ${\cos }^{-1}\left(x\right)$ to ${\sin }^{-1}\left(x\right)$?

Answer 1

Transforming the given inverse trigonometric function:

Let $\cos \left(\theta \right)=x$

$\Rightarrow$ $\theta ={\cos }^{-1}\left(x\right)...\left(1\right)$

Using the identity: ${\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1$

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

From $\left(1\right)$, we have,

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

Hence, we get the relation ${\cos }^{-1}\left(x\right)={\sin }^{-1}\left(\sqrt{1-x^2}\right)$.