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

1 Answer
Jan 6, 2016

Answer:

#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)#


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)#