How do you solve tan(x + y) = (tan x + tan y) / (1 - tan x tan y)?

We have to prove tan(x+ y) = (tan x + tan y) / (1 – tan x tan y)

Proof

We will use the relations that tan x = sin x / cos x.

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

cos (x+ y) = cos x * cos y – sin x * sin y

tan (x+ y) = sin (x + y) / cos (x + y)

=> [sin x * cos y + cos x * sin y] / [cos x * cos y – sin x * sin y]

divide all the terms yy cos x * cos y

=>[ (sin x * cos y)/(cos x * cos y)+ (cos x * sin y)/(cos x * cos y)] / [(cos x * cos y)/(cos x * cos y) – (sin x * sin y)/(cos x * cos y)]

=> [(sin x / cos x) + (sin y / cos y)]/[ 1 – (sin x / cos x)*(sin y/ cos y)]

=> (tan x + tan y) / (1 – tan x * tan y)

=> RHS

Hence proved

Therefore we hxve tan (x + y) = (tan x + tan y) / (1 – tan x * tan y)