Problem 38

Question

Verify each identity. (a) $\arcsin (-x)=-\arcsin x, \quad|x| \leq 1$ (b) $\arccos (-x)=\pi-\arccos x, \quad|x| \leq 1$

Step-by-Step Solution

Verified

Answer

For $|x| \leq 1$, $\arcsin(-x)=-\arcsin x$ and $\arccos(-x) = \pi-\arccos x$ are both valid mathematical identities.

1Step 1: Verifying Identity for $\arcsin (-x)=-\arcsin x$

Firstly, consider the left-hand side, $\arcsin(-x)$. If $\sin y = -x$, then $-\sin y = x$. As sine is an odd function, meaning $\sin(-y) = -\sin(y)$, so $\sin(-y) = x$, meaning $-y = \arcsin x$. Hence, $\arcsin(-x) = -\arcsin x$ is validated for any $x$ such that $|x| \leq 1$.

2Step 2: Verifying Identity for $\arccos(-x) = \pi-\arccos x$

For the second identity, start with the right-hand side, $\pi - \arccos x$. If $\cos y = x$, then we have $\cos(\pi - y) = -x$. As cos is an even function, meaning $\cos(-y) = \cos(y)$, and here $y = \pi - \arccos x$, which leads us $\cos(-y) = -x $, this means $-y = \arccos(-x)$. Hence, $\pi - \arccos x = \arccos(-x)$ is validated for any $x$ such that $|x| \leq 1$.

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