

We know that
tanx = sinx/cosx
Now, differentiate w.r.t. x, we get
d(tanx)/dx = d(sinx/cosx)/dx
= {cosx*cosx - sinx*(-sinx)}/cos2 x
= {cosx * cosx + sinx*sinx}/cos2 x
= (cos2 x + sin2 x)/cos2 x
= 1/cos2 x {since cos2 x + sin2 x = 1 }
= sec2 x
So, d(tanx)/dx = sec2 x
