The maximum value of sin x*cos x - tan x has to be determined.

Consider the first term in the expression sin x*cos x, the lowest value of this is 0 which is the case when either of sin x or cos x is 0 and the maximum value is 1/2. The value of the given expression increases as the value of tan x decreases. tan x = sin x/cos x, if sin x is negative, the value of tan x decreases as cos x approaches 0. When cos x is 0 and sin x is negative, tan x takes the value `-oo` . At all points where this is the case, the value of the given expression is `oo` .

You could determine the values of x where sin x*cos x - tan x equals `oo` but this is not asked for in the problem.

The following graph illustrates the change in value of sin x*cos x - tan x with x. notice the points where the value increases to infinity.