TITLE: why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$
QUESTION [5 upvotes]: How do i  find  the  When $x$ is  ranging from $-1$ to $1$?
I want know why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$
I have already tried inverse-function.
thanks.

REPLY [6 votes]: There are a couple of ways of doing this; here's a short, simple geometric way. First, we draw a right triangle and label the vertices $A,B,C$ and the side lengths $a,b,c$.

We have$$\begin{align*} & \sin\theta=\frac bc\implies\theta=\arcsin\frac bc\\ & \cos\varphi=\frac bc\implies\varphi=\arccos\frac bc\end{align*}$$Obviously, the sum of $\theta$ and $\varphi$ is $90^{\circ}$. Therefore, it follows that$$\arcsin\frac bc+\arccos\frac bc=\frac {\pi}2$$