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Prove that : sintheta/(cos(3theta)) + ...

Prove that : `sintheta/(cos(3theta)) + (sin3theta ) /(cos9theta) + (sin 9theta) /(cos27theta) =1/2( tan27theta-tantheta)`

Text Solution

Verified by Experts

`tan3theta - tan theta =( sin3theta)/(cos3 theta) - sin theta/cos theta`
`= (sin3thetacostheta - sinthetacos3theta)/(cos3thetacos theta)`
`=sin(3theta-theta)/(cos3thetacos theta)`
`=sin(2theta)/(cos3thetacos theta)`
`=(2sinthetacostheta)/(cos3thetacos theta) = (2sintheta)/(cos3theta)`
`:. 1/2(tan3theta - tan theta) = (sintheta)/(cos3theta)`
Similarly,
`1/2(tan9theta - tan 3theta) = (sin3theta)/(cos9theta)`
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