Let's start from a definition of the functions mentioned in the problem.

(1) Evaluation of #tan^-1(0)#

By definition, #tan(phi)=sin(phi)/cos(phi)#.

Therefore, #tan^-1(phi)=cos(phi)/sin(phi)#

By definition, #cos(phi)# is an abscissa of a point on a unit circle that is an endpoint of a radius at angle #phi# to the X-axis. Also by definition, #sin(phi)# is an ordinate of this point.

If an angle #phi=0#, the abscissa #cos(0)# of an endpoint of a corresponding radius equals to #1#, while it's ordinate #sin(0)# equals to #0#.

As we see, the denominator of the expression for

#tan^-1(0)=cos(0)/sin(0)# equals to #0#, which means that #tan^-1(0)# is UNDEFINED.

(2) Evaluation of #csc^-1(2)#

By definition, #csc(phi)=1/sin(phi)#

Therefore, #csc^-1(phi)=sin(phi)#

As we stated above, by definition, #sin(phi)# is an ordinate of the endpoint of a radius that forms an angle #phi# with the X-axis.

If an angle #phi=2# (that is, #2# radians), the abscissa of the endpoint of a corresponding radius is, approximately, #0.91#. That is the value of #csc^-1(2)#.