Inverse Trigonometric Functions

In this section, we shall prove some important properties of inverse trigonometric functions. It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions. In fact, they will be valid only for some values of x for which inverse trigonometric functions are defined. We will not go into the details of these values of x in the domain as this discussion goes beyond the scope of this textbook.

Let us recall that if \(y = sin^{-1}x\), then \(x = sin y\) and if \(x = sin y\), then \(y = sin^{-1}x\). This is equivalent to \(sin(sin^{-1}x) = x, x \in [-1, 1]\) and \(sin^{-1}(sin x) = x, x \in [-\frac{\pi}{2}, \frac{\pi}{2}]\) For suitable values of domain similar results follow for remaining trigonometric functions.