Question

Show that${\sin }^{-1}\left(v\right)+{\cos }^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2}$

Short answer

The equation ${\sin }^{-1}\left(v\right)+{\cos }^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2}$can be obtained as

$$\begin{gathered} x={\sin }^{-1}\left(v\right) \\ y={\cos }^{-1}\left(v\right) \\ so\sin \left(x\right)=v \\ \cos \left(y\right)=v \\ \sin \left(x\right)=\cos \left(y\right) \\ \sin \left(x\right)=\sin \left(\frac{\mathrm{\pi }}{2}-y\right) \\ x=\frac{\mathrm{\pi }}{2}-y \\ x+y=\frac{\mathrm{\pi }}{2} \\ {\sin }^{-1}\left(v\right)+{\cos }^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Step-by-step solution

Step 1. Given data

The given equation for verification is

${\sin }^{-1}\left(v\right)+{\cos }^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2}$

Step 2. Considering variables

Consider

$$\begin{gathered} x={\sin }^{-1}\left(v\right) \\ y={\cos }^{-1}\left(v\right) \end{gathered}$$

Change inverse trigonometric into normal trigonometric function

$$\begin{gathered} \sin \left(x\right)=v \\ \cos \left(y\right)=v \end{gathered}$$

so localid="1646412059960" $\sin \left(x\right)=\cos \left(y\right)$

Step 3. proof

Use formula $\cos \left(n\right)=\sin (\frac{\mathrm{\pi }}{2}-n)$

localid="1646412107273" $$\begin{gathered} \sin \left(x\right)=\cos \left(y\right) \\ \sin \left(x\right)=\sin \left(\frac{\mathrm{\pi }}{2}-y\right) \\ x=\frac{\mathrm{\pi }}{2}-y \\ x+y=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Substitute expression of xand y

$$\begin{gathered} x+y=\frac{\mathrm{\pi }}{2} \\ {\sin }^{-1}\left(v\right)+{\cos }^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2} \end{gathered}$$