Although this was in the "Differentiating Trigonometric Functions" section, it looks like you just want to evaluate or simplify what you provided.
y=3sec^-1(sqrt(2))-4csc^-1(sqrt(2))+5cot^-1(2)
FIRST TERM: sec^-1(sqrt(2))
If x=sec^-1(sqrt(2)), then sec(x)=sqrt(2) or 1/cos(x)=sqrt(2).
This is the same as saying
cos(x)=1/sqrt(2)=sqrt(2)/2, which happens when x=pi/4
SECOND TERM: csc^-1(sqrt(2))
If x=csc^-1(sqrt(2)), then csc(x)=sqrt(2)
Identically, 1/sin(x)=sqrt(2) or sin(x)=1/sqrt(2)=sqrt(2)/2, which also happens at x=pi/4
THIRD TERM: cot^-1(2)
If x=cot^-1(2), then cot(x)=2 or 1/tan(x)=2 or tan(x)=1/2. Because the function y=tan(x) repeats periodically (see graph), you actually get an infinite number of possible x values. So you can't really say what cot^-1(2) is.
graph{y=tan(x) [-20,20,-0.5,1]}
Most calculators assume you meant the middle one though and will give you an answer cot^-1(2)~~0.4636
ANSWER
Plugging those values in, you get
y=3(pi/4)-4(pi/4)+5cot^-1(2)
y=5cot^-1(2)-pi/4
A calculator will deceptively say y~~1.5326
