I am trying to find the range and domain of the function :
$\frac{(\sin^{-1}x+\tan^{-1}x-\cos^{-1}x-\cot^{-1}x)}{(\sin^{-1}x+\tan^{-1}x+\cos^{-1}x+\cot^{-1}x)}$
My attempt:
As $\sin^{-1}x$ has the smallest domain among the functions involved, the domain of the entire function should be $[-1,1]$. Now, I know that individual range of the functions and I am also aware of general properties of inverse trigonometric functions, but I am unable to find the range of this one.
All help is appreciated.
You have correctly determined the domain to be restricted to $[-1,1]$.
For the range, we note that denominator of the given function (call $f$) is a constant, that is $\pi$ and numerator is strictly increasing. So the range is $[f(-1),f(1)]$.