Question #1e28e

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Answer 1 · 2017-03-15T08:47:49

see below

#tan3theta=tan(2theta+theta)#

#using the compound angle identity

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

#tan3theta=(tan2theta+tantheta)/(1-tan2thetatantheta)#

using the same identity on #" "tan2theta#

#tan3theta=((tantheta+tantheta)/(1-tanthetatantheta)+tantheta)/(1-(tantheta+tantheta)/(1-tanthetatantheta)tantheta)#

now to tidy up

#tan3theta=((2tantheta)/(1-tan^2theta)+tantheta)/(1-(2tantheta)/(1-tan^2theta)tantheta)#

#tan3theta=((2tantheta+tantheta-tan^3theta)/cancel((1-tan^2theta)))/((1-tan^2theta-2tan^2theta)/cancel((1-tan^2theta)))#

#tan3theta=((2tantheta+tantheta-tan^3theta))/((1-tan^2theta- 2tan^2theta)#

giving

#tan3theta=(3tantheta-tan^3theta)/(1-3tan^2theta)#