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Prove that: (sinx)/(cos3x)+(sin3x)/(cos...

Prove that: `(sinx)/(cos3x)+(sin3x)/(cos9x)+(sin9x)/(cos27 x)=(1/2)(tan27 x-tanx)`

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Given,
`=>(sinx)/(cos3x)+(sin3x)/(cos9x)+(sin9x)/(cos27 x)`
Now,
multiplying and dividing by 2
`=(1/2)​[2sinx​)/(cos3x)+[2sin3x)/(cos9x)+(2sin9x)/(cos27x)]`
multiplying and dividing by `cosx,cos9x,cos27`
`=(1/2)​[(sin2x)/(cos3xcosx)+(sin6x​)/(cos9xcos3x)+(sin18x)/(cos27xcos9x)​]`
`=(1/2)​[(sin3xcosx−cos3xsinx)/(cos3xcosx)​]+(1/2)​[(cos3xsin9x−cos9xsin3x)/(cos27xcos9x)​]+(1/2)​[(sin27xcos9x−cos27xsin9x)/(cos27xcos9x)​]`
...
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