I don't really understand why we sketch the graphs of **cos^-1(cosx)**, **cos(cos^-1(x))** , **sin^-1(sinx)** and **sin(sin^-1(x)) **the way they are in the textbook. I would appreciate it if someone could explain the reason behind sketching them that way!

Thanks in advance!

**cos(cos^-1(x)), sin(sin^-1(x)) **should be easy. Use the theorem

**f(f^-1(x)) = x.**
sin^-1(sinx), cos^-1(cosx) you need to examine by inspection to sketch and get the pattern. Use the theorem

** f^-1(f(x)) = x **but this only holds for the domain of the sin^-1 / cos^-1

As an excerise try to prove the theorems:

**1. f(f^-1(x)) = x**

2. f^-1(f(x)) = x
Observe for both theorems the argument of the function, in simple terms that is the input the output so long the domain constraint is satisfied.

Hint: For proving the theorems, the functions are defined differently in 1/2.