Let's start from a definition of the functions mentioned in the problem.
(1) Evaluation of tan−1(0)
By definition, tan(ϕ)=sin(ϕ)cos(ϕ).
Therefore, tan−1(ϕ)=cos(ϕ)sin(ϕ)
By definition, cos(ϕ) is an abscissa of a point on a unit circle that is an endpoint of a radius at angle ϕ to the X-axis. Also by definition, sin(ϕ) is an ordinate of this point.
If an angle ϕ=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(ϕ)=1sin(ϕ)
Therefore, csc−1(ϕ)=sin(ϕ)
As we stated above, by definition, sin(ϕ) is an ordinate of the endpoint of a radius that forms an angle ϕ with the X-axis.
If an angle ϕ=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).
