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Prove that sinx/cos(3x)+sin(3x)/cos(9x)+...

Prove that `sinx/cos(3x)+sin(3x)/cos(9x)+sin(9x)/cos(27x)=1/2 (tan27x-tanx)`

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Given,`sinx/cos(3x)+sin(3x)/cos(9x)+sin(9x)/cos(27x)`
multiplying and dividing by 2
=1/2 xx `(2sinx)/cos(3x)+(2sin(3x))/cos(9x)+(2sin(9x))/cos(27x)`
multiplying and dividing by cosx,cos9x,cos27 respectively and simplifying
=`1/2​xx[(sin2x)/(cos3xcosx)+(sin6x​)/(cos9xcos3x)+(sin18x)/(cos27xcos9x​)]`
Use, sin(A−B)=sinAcosB−cosAsinB
=`1/2​[(cosx−cos3xsinx)/(cos3xcosxsin3x)​]+1/2​[(sin9xcos3x−cos9xsin3x​)/(cos27xcos9x)]+1/2​[(sin27xcos9x−cos27xsin9x​)/(cos27xcos9x)]`
=`1/2[tan27x−tan9x+tan9x−tan3x+tan3x−tanx]=1/2​[tan27x−tanx]`
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