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) =

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

sin (x - y) =
y*cos x

cos (x - y) =
y

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

We'll force factor , both
numerator and denominator:

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

We'll reduce like
terms:

tan(x-y) =
y)

The requested formula for tan(x - y) is:

.