sec^2 x - sec^2 y = tan^2 x - tan^2 y

We notice the differences of squares from both
sides:

(sec x)^2 - (sec y)^2 = (sec x - sec y)(sec x + sec
y)

(tan x)^2 - (tan y)^2 = (tan x - tan y)(tan x + tan
y)

We also know that sec x = 1/cos x and sec y = 1/cos
y

sec x - sec y = 1/cos x - 1/cos
y

sec x - sec y = (cos y - cos x)/cos y*cos
x

cos y - cos x =
2sin[(x+y)/2]sin[(x-y)/2]

sec x + sec y = 1/cos x + 1/cos
y

sec x + sec y = (cos y + cos x)/cos y*cos
x

cos y + cos x =
2cos[(x+y)/2]cos[(x-y)/2]

(sec x - sec y)(sec x + sec y) =
sin 2[(x+y)/2]*sin 2[(x-y)/2]/(cos y*cos x)^2

We'll
simplify and we'll get:

(sec x - sec y)(sec x
+ sec y) = sin (x+y)*sin (x-y)/(cos y*cos
x)^2

We'll work now on the right side of the
equal:

tan x = sin x/cos x

tan
y = sin y/cos y

tan x - tan y = (sin x*cosy -
siny*cosx)/cos x*cos y

tan x - tan y = sin (x-y)/cos x*cos
y

tan x + tan y = sin (x+y)/cos x*cos
y

(tan x - tan y)(tan x + tan y) = sin
(x-y)*sin (x-y)/(cos x*cos y)^2

We notice
that we've get the same expression both sides, so the identity is
verified.