f'=1/(cosx)^2
f''=2cosxsinx/(cosx)^4=2sinx/(cosx)^3
f'''={2(cosx)^4+6(sinx)^2(cosx)^2}/(cosx)^6

f'(0)=1,f''(0)=0,f'''(0)=2
tanx=x +2x^3/3! +o(x^3) =x +x^3/3 +o(x^3)