Show how to get the formula for tan(x-y)?

We'll compute the formula of tangent of a difference of
two angles, using the information that tangent function can be written as a
fraction:

tan(x-y) =
`[sin(x-y)]/[cos(x-y)]`

We'll recall the formula for
sin(x-y) and cos(x-y):

sin (x - y) = `sin x*cos y - sin
y*cos x`

cos (x - y) = `cos x*cos y + sin x*sin
y`

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

We'll force factor `cos x*cos y` , both
numerator and denominator:

tan(x-y) = `[cos x*cos y(tan x -
tan y)]/[cos x*cos y(1 + tan x*tan y)]`

We'll reduce like
terms:

tan(x-y) = `(tan x - tan y)/(1 + tan x*tan
y)`

The requested formula for tan(x - y) is:
`tan (x - y) = (tan x - tan y)/(1 + tan x*tan y)`
.