Problem is csc-1x in terms of tan-1. And i cant really get a grip of it.

Tried it in two ways which are both wrong.

First attempt csc-1(x)=tan-1(y)=z

Thus csc(z)=x=1/sin(z) and sin(z)=1/x

tan(z)=y=sin(z)/cos(z)=sin(z)/(1-sin^2(z))^1/2) At this point i think it all gets messed up. substituting in the values for sinz=1/x gives me

y=1/((x^2-1))^1/2.

Thus tan^-1(1/((x^2-1))^1/2)=csc-1(x)=z. Which is incorrect.

And when things get wrong you try other things so here is my other attempt

csc-1(x)=sin-1(1/x)=pi/2 - cos-1(1/x))=tan-1(y)=z. Thus

csc(z)=1/sin(z)=1/(pi/2 - cos(z))=x

And tan(z)=y=sin(z)/cos(z). Substituting the values gives me y=2/(pi*x - 2). Performing the inverse and I end up with tan-1(2/(pi*x - 2))=z=csc-1(x).

Again the solution is wrong.

Correct solution is pi/2− sgnxtan-1((x^2-1)^1/2)

And i dont know how to get there. Thankful if anyone can point out where im thinking wrong so I can correct it for future problems.