It's based on how they are defined and the nature of the graphs of sine, cosine, and tangent (I'll assume you are familiar with their graphs in what follows).
For example, x=f^(-1}(y)=sin^{-1}(y) is defined to be the inverse function of y=f(x)=sin(x) for -pi/2\leq x\leq pi/2. Since y=f(x)=sin(x) is continuous and y->1 as x->\frac{pi}{2}^{-} (the minus sign to the right of the number indicates the number is being approached "from the left"), it follows that x=f^{-1}(y)=sin^{-1}(y)->pi/2 as y->1^{-}. Similarly, x=f^{-1}(y)=sin^{-1}(y)->-\pi/2 as y->-1^{+}.
Since x=g^{-1}(y)=cos^{-1}(y) is defined to be the inverse function of y=g(x)=cos(x) for 0\leq x\leq pi and both functions are continuous, it follows that x=g^{-1}(y)=cos^{-1}(y)->0 as y->1^{-} and x=g^{-1}(y)=cos^{-1}(y)->pi as y->-1^{+}.
Since x=h^{-1}(y)=tan^{-1}(y) is defined to be the inverse function of y=h(x)=tan(x) for -pi/2 < x < pi/2 and both functions are continuous, it follows that x=h^{-1}(y)=tan^{-1}(y)->pi/2 as y->+\infty and x=h^{-1}(y)=tan^{-1}(y)->-pi/2 as y->-\infty.
