Question

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

Answer 1

`tan(x+y)=(tanx+tany)/(1-tanxtany)`

Explanation:

This can be expanded through the tangent angle addition formula:

`tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)`

Thus,

`tan(x+y)=(tanx+tany)/(1-tanxtany)`

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

`sin(alpha+beta)=sinalphacosbeta+cosalphasinbeta`
`cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta`

Since `tanx=sinx/cosx`,

`tan(alpha+beta)=sin(alpha+beta)/cos(alpha+beta)=(sinalphacosbeta+cosalphasinbeta)/(cosalphacosbeta-sinalphasinbeta)`

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

`tan(alpha+beta)=((sinalphacosbeta+cosalphasinbeta)/(cosalphacosbeta))/((cosalphacosbeta-sinalphasinbeta)/(cosalphacosbeta))=(sinalpha/cosalpha(cosbeta/cosbeta)+sinbeta/cosbeta(cosalpha/cosalpha))/(cosalpha/cosalpha(cosbeta/cosbeta)-sinalpha/cosalpha(sinbeta/cosbeta))`

Final round of simplification yields:

`tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)`