By writing tan x as sin x cos x , use the quotient rule to show that d dx ðtan xÞ ¼ sec2 x .

First write tanx as sinx/cosx as it is always helpful to use what additional information the question gives you. It says we must use the quotient rule to calculate the result so it is also a good idea to write out the quotient rule so we know what values we need to work out. Quotient rule: dy/dx = (u'v-v'u)/v^2 where u=sinx and v=cosx. So we are required to work out u' and v'. Once we have done this, we substitute all the values into the quotient rule. Then using the identity sin^2(x)+cos^2(x)=1 we can see that dy/dx=1/cos^2(x). Now 1/cosx=secx, thus dy/dx=sec^2(x).