Is arcsin(x) = csc(x) true?

1 Answer
Oct 21, 2015

No. This is confusing #sin^-1(x)# with #(sin(x))^-1#.

Explanation:

#arcsin(x) = sin^-1(x)# is the inverse function of the function #sin(x)#

That is:

If #x in (-pi/2, pi/2)#, then #arcsin(sin(x)) = x#

If #x in [-1, 1]# then #sin(arcsin(x)) = x#

On the other hand:

#csc(x) = (sin(x))^(-1) = 1/sin(x)# is the reciprocal of the #sin# function.

I think some of the blame for this confusion has to lie with the common convention of writing #sin^2(x)# to mean #sin(x)^2#. So when you have #csc(x) = 1/sin(x) = sin(x)^(-1)# you might think that we would also write that as #sin^(-1)(x)#, but that's reserved for #arcsin(x)#.