Prove that
`tan alpha+2 tan 2 alpha+2^(2)tan^(2)alpha+..............+2^(n-1)tan2^(n-1)alpha+2^(@)cot2^(@)alpha=cotalpha`
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We know `tan theta=cot theta-2 cot 2 theta`.
Putting `theta=alpha,2alpha,2^(2)alpha,....................` in (i), we get
`tan alpha=(cot alpha-2 cot 2alpha)`
`2(tan2alpha)=2(cot 2 alpha-2 cot 2 ^(2)alpha)`
`2^(2)(tan2^(2)alpha)=2^(2)(cot2^(2)alpha-2cot2^(3)alpha)`
`2^(n-1)(tan2^(n-1)alpha)=2^(n-1)(cot2^(n-1)alpha-2cos2^(n)alpha)`
Adding,
`tan alpha+2tan 2alpha+2^(2)tan^(2)alpha+..........+2^(n-1)tan2^(n-1)alpha=cotalpha-2^(n)cot2^(n)alpha`
`therefore tan alpha+2 tan 2 alpha+2^(2)tan^(2)alpha+..........+2^( n-1)alpha+2^(n)cot2^(n)alpha=cotalpha`.