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).
