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

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lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

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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)

devide the numerator and denominator by cos x.cos y

= (sin x/cos x - sin y/cos y) / (1 + sin x.sin y/cos x.cos y)

= (tan x - tan y)/(1 + tan x.tan y)

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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)` .

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