why inverse(sin(-x)) equivalent with -(inverse(sin(x)))

Question

anonymous · Answer

because both the inside function and the outside one are odd

ash2326 · Answer

$$\sin^{-1} (-x)=-\sin^{-1} x$$
because $\sin^{-1} x$ is an odd function

anonymous · Answer

please tell me, what do you mean they both odd function?

anonymous · Answer

$$\sin(-x)=-\sin(x)$$ and 
$$\sin^{-1}(-x)=-\sin{-1}(x)$$ 
therefore 
$$\sin^{-1}(\sin(-x))=\sin^{-1}(-sin(x))=-\sin^{-1}(\sin(x))$$

ash2326 · Answer

if 
f(-x)=-f(x) then the  function is odd

anonymous · Answer

definition :

$$f \text { is odd if } \f(-x)=-f(x)$$

anonymous · Answer

thanks , tan is odd function to right?

ash2326 · Answer

yeah tan x is odd

anonymous · Answer

sorry, what about cos?

ash2326 · Answer

cos x is even
$$cos (x)=cos(-x)$$

anonymous · Answer

oh I see what do you mean. interesting..

anonymous · Answer

what about cos$$\sec ^{-1}(-x)=\pi+\sec^(-1)$$

turingtest · Answer

anonymous · Answer

$$\sec^−1(−x)=π+\sec ^{-1}$$