cos2x

dy

dx

+ y = tan x
⇒

dy

dx

+ sec2 x.y = sec2 x tan x
This equation is in the form of

dy

dx

+ py = Q
here p = sec2 x and Q = sec2 x tan x
Integrating Factor , I.F = e∫pdx = e∫sec2xdx = etanx
The general solution can be given by
y (I.F.) = ∫ (Q × I.F.) dx + C ... (1)
Let tanx = t
⇒

d

dx

(tan x) =

dt

dx

⇒ sec2 x =

dt

dx

⇒ sec2 x dx = dt
Therefore, equation 1 becomes :
y.etanx = ∫ (et.t) dt
⇒ y . etanx = ∫ (et.t) dt + C
⇒ y . etanx = t . ∫ et dt - ∫ (

d

dt

(t).∫et)dt + C
⇒ y . etanx = t.et−∫etdt + C
⇒ y . etanx = t.et−et + C
⇒ y . etanx = (t - 1) et + C
⇒ y . etanx = (tan x - 1) etanx + C
⇒ y = (tan x - 1) + Ce−tanx , where C is an arbitary constant