cos2x

dy

dx

+ y = tan x

dy

dx

+sec2x.y = tan x . sec2 x
It is a linear differential equation of the first order.
Comparing it with

dy

dx

+ Py = Q, we get
P = sec2 x and Q = tan x. sec2 x
Integration factor = e∫Pdx = e∫sec2xdx = etanx
The solution of the given linear differential equation is given as:
y etanx = ∫ tan x . sec2 x . etanx dx + C
Let tan x = t ⇒ sec2 x dx = dt
yet = ∫ t . et . dt + C
yet = t.et - ∫ 1.et dt + C
yet = t.et−et + C
yet = et (t - 1) + C
yetanx = etanx (tan x - 1) + C
y = tan x - 1 + Ce−tanx