Which are equal?

Calculus Level 4

Consider the following functions:

I. $a(x) =$ $\cosh^{-1}(\sec(x))$

II. $b(x) =$ $\tanh^{-1}(\sin(x))$

III. $c(x) =$ $\ln(\csc(x)+\cot(x))$

IV. $d(x) =$ $\ln(\sec(x)+\tan(x))$

Which of these functions are always equivalent?

Note: Let each of the domains of these functions be the most inclusive domain.

II, III, IV I, II, III I, II, IV I, III, IV None of them are equivalent.

1 solution

Dallin Richards
Mar 25, 2017

I, II, and IV are each anti-derivatives of $\sec(x)$ which happen to not differ by a constant.