Mathematics

Problem

(a) The derivatives of

\tan x

and

\cot x

are

\sec^{2}x

and

-\csc^{2}x

respectively. Explain, without calculating any numerical value, why the derivative of

\cot x

is always negative while that of

\tan x

is always positive on the interval

\left(0,\dfrac{\pi}{2}\right)

. (b) Using part (a), discuss the monotonic behaviour of each of the functions

y=\tan x

and

y=\cot x

\left(0,\dfrac{\pi}{2}\right)

, stating whether each function is increasing or decreasing and giving a brief justification.