How would I write $\sin^{-1}(a+b)$ in terms of $\sin^{-1}(a)$ and $\sin^{-1}(b)$? Assume that $\sin^{-1}(a+b)$ exists. Is it possible because I know you can easily write out $\sin^{-1}(a) + \sin^{-1}(b)$
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1There's no simple expression for the general $a,b$ as far as I know – Yuriy S Jan 23 '18 at 23:26
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Please, if you are ok, you can accept the answer and set it as solved. Thanks! – user Jan 28 '18 at 07:51
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Since $\arcsin x$ is odd what we can say is that
$$\arcsin a + \arcsin (-a)= \arcsin (a-a)=\arcsin 0 =0 $$
but there are not simple expression for the general $a$ and $b$.
One reason for this is that $\arcsin x$ is defined for $x\in[-1,1]$ thus if we consider, for example, $a=1$ and $b=\frac12$ we can define $\arcsin a$ and $\arcsin b$ but $\arcsin(a+b)$ is meaningless.
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There are ways that the expression can be undefined if $|a+b|>1$ – Ralph Tarantino Jan 23 '18 at 23:58
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@RalphTarantino We could also take $\arcsin(100+(-99))=\frac{\pi}{2}$ but $\arcsin(100)$ and $\arcsin(-99)$ are not defined. This is not obviously a proof but can give an idea for the fact that is not simple to define such relations for this function. – user Jan 24 '18 at 00:08