Answer
x = $\frac{3\pi}{4}$ and x = $\frac{7\pi}{4}$
Work Step by Step
The rate of change of a function is the same as its derivative. Thus, this question is asking us when the derivative of f(x) and g(x) are equal.
f(x) = sec(x)
g(x) = csc(x)
The derivative of sec(x) = secxtanx
The derivative of csc(x) = -cscxcotx
Setting these equal to eachother, we get
sec(x)tan(x) = -csc(x)cot(x)
$\frac{sec(x)tan(x)}{csc(x)cot(x)}$ = -1
Taking the derivative of this, we get
$\frac{\frac{1}{cos(x)}\times\frac{sin(x)}{cos(x)}}{\frac{1}{sin(x)}\times\frac{cos(x)}{sin(x)}}$ = -1
Simplifying this, we get $\frac{(sin(x))^{3}}{(cos(x))^{3}}$ = -1.
$\frac{sin(x)}{cos(x)}$ = tan(x), thus $\frac{(sin(x))^{3}}{(cos(x))^{3}}$ = (tan(x))$^{3}$ = -1.
tan(x) = $\sqrt[3] -1$ = -1
The only x values for which this statement is true on the interval (0, 2$\pi$] are x = $\frac{3\pi}{4}$ and x = $\frac{7\pi}{4}$
