Find a formula relating $\arcsin (x)$ and $\arccos (x)$

Question

From the formula $\sin(\frac{\pi}{2}-x)=\cos x$, find a formula relating $\arcsin (x)$ and $\arccos (x)$.

I have no idea where to start.

Answer 1

Notice, let $$\cos \theta =x\implies \theta=\cos^{-1}(x)$$

Now, we know that $$\sin\left(\frac{\pi}{2}-\theta\right)=\cos \theta$$ setting $\cos \theta=x$, we get $$\sin\left(\frac{\pi}{2}-\theta\right)=x$$ $$\frac{\pi}{2}-\theta=\sin^{-1}(x)$$ setting $\theta=\cos^{-1}(x)$, we get $$\frac{\pi}{2}-\cos^{-1}(x)=\sin^{-1}(x)$$ $$\color{red}{\sin^{-1}(x)+\cos^{-1}(x)=\frac{\pi}{2}}$$

Answer 2

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