How do you simplify tan(x+y) to trigonometric functions of x and y?

1 Answer
Jan 6, 2016

tan(x+y)=tanx+tany1tanxtany

Explanation:

This can be expanded through the tangent angle addition formula:

tan(α+β)=tanα+tanβ1tanαtanβ

Thus,

tan(x+y)=tanx+tany1tanxtany


The tangent addition formula can be found using the sine and cosine angle addition formulas.

sin(α+β)=sinαcosβ+cosαsinβ
cos(α+β)=cosαcosβsinαsinβ

Since tanx=sinxcosx,

tan(α+β)=sin(α+β)cos(α+β)=sinαcosβ+cosαsinβcosαcosβsinαsinβ

This can be written in terms of tangent by dividing both the numerator and denominator by cosαcosβ.

tan(α+β)=sinαcosβ+cosαsinβcosαcosβcosαcosβsinαsinβcosαcosβ=sinαcosα(cosβcosβ)+sinβcosβ(cosαcosα)cosαcosα(cosβcosβ)sinαcosα(sinβcosβ)

Final round of simplification yields:

tan(α+β)=tanα+tanβ1tanαtanβ