Prove that the d(tan(x))/dx is equal to sec^2(x).

You can express tan(x) as sin(x)/cos(x). Therefore, tan(x)= sin(x)/ cos(x)The quotient rule can be applied here as there is a function of x in the numerator and denominator.Quotient Rule: (v*(du/dx) - u*(dv/dx))/v²Let u =sin(x) and v=cos(x) and hence (du/dx)= cos(x) and (dv/dx)= -sin(x).Therefore:d(tan(x))/dx= (cos(x)cos(x))-(sin(x)(-sin(x))/(cos²(x))=(cos²(x)+sin²(x))/(cos²(x))Using the trig identity, cos²(x)+sin²(x)=1, the numerator of the fraction can be tidied and heavily simplified.d(tan(x))/dx= 1/(cos²(x))As 1/(cos(x)) is equal to sec(x), 1/(cos²(x)) is equal to sec²(x).