Simplify cos(sin^-1(x)) | Simplify cos(arc sin x)

 The value of cos(sin^-1(x)) or cos(arc sin x) is equal to $\sqrt{1-x^2}$. In this post, we will simplify the expression $\cos(\sin^{-1}(x))$.

We note that both $\cos(\sin^{-1}(x))$ and $\cos(arc \sin x)$ have the same value. Thus to find out the value of $\cos(\sin^{-1}(x))$ or $\cos(arc \sin x)$, let us put

$\theta=\sin^{{-1}} x$ $\cdots (\star)$

 

As $\theta=\sin^{{-1}} x$ and we have to simplify $\cos(\sin^{-1}(x))$, we need to find out the value of $\cos \theta$. From the above equation, we get that

 

$x=\sin \theta$, that is,

 

$\sin \theta=x$ $\cdots (\star \star)$

 

Now, we will apply the Pythagorean trigonometric identity $\sin^2 \theta+\cos^2 \theta =1$. From this identity, we have that

 

$\cos^2\theta = 1-\sin^2 \theta$

 

$\Rightarrow \cos \theta = \sqrt{1-\sin^2 \theta}$

 

$=\sqrt{1-x^2}$ [ putting the value of $\sin \theta$ from $(\star \star)$ ]

$\therefore \cos \theta =\sqrt{1-x^2}$

 

Now, put the value of $\theta$ from $(\star)$

So, $\cos(\sin^{-1}(x))=\sqrt{1-x^2}$. Thus, the formula of $\cos(\sin^{-1}(x))$ is $\sqrt{1-x^2}$.