Question

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

Short answer

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

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

Step-by-step solution

Step 1. Given data 

The given equation for verification is

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

Step 2. Proof

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

Use Equation $1$in the given equation

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

as $\sin \left(\frac{\mathrm{\pi }}{2}\right)=1$

So$\sin ({\sin }^{-1}\left(v\right)+{\cos }^{-1}(v\left)\right)=1$