Prove that #{sin(theta)+sin(3theta)+sin(5theta)+sin(7theta)}/{cos(theta)+cos(3theta)+cos(5theta)+cos(7theta)}=tan(4theta)# ?

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Answer 1 · 2017-11-13T13:36:33

Please refer to a Proof in the Explanation.

We have,
#(sintheta+sin3theta+sin5theta+sin7theta)/(costheta+cos3theta+cos5theta+cos7theta),#

#={(sin7theta+sintheta)+(sin5theta+sin3theta)}/{(cos7theta+costheta)+(cos5theta+cos3theta)},#

#={(2sin4thetacos3theta)+(2sin4thetacostheta)}/{(2cos4thetacos3theta)+(2cos4thetacostheta)},#

#={cancel2sin4thetacancel((cos3theta+costheta))}/{cancel2cos4thetacancel((cos3theta+costheta))},#

#=(sin4theta)/(cos4theta),#

#=tan4theta.#

Hence, the Proof.