Prove that
`tan alpha+2 tan 2 alpha+2^(2)tan2^(2)alpha+..............+2^(n-1)tan2^(n-1)alpha+2^(n)cot2^nalpha=cotalpha`

Explanation

We know `tantheta=cottheta-2cot2theta`…..(i)
Putting `theta=alpha,2alpha,2^(2)alpha,…………….in (ii)`, we get
`tanalpha = (cotalpha-2cot2alpha)`
`2(tan2alpha)= 2(cot2alpha-2cot2^(2)alpha)`
`2^(2)(tan2^(2)alpha) = 2^(2)(cot2^(2)alpha-2cot2^(2)alpha)`
Adding,
`tanalpha+2tan2alpha+2^(2)tan^(2)alpha+...........+2^(n-1)tan2^(n-1)alpha=cotalpha-2^(n)cot2^(n)alpha`
`therefore tanalpha+2tan2alpha+2^(2)tan^(2)alpha+............+2^(n-1)tan2^(n-1)alpha=cotalpha-2^(n)alpha`
`therefore tanalpha+2tan2alpha+2^(2)tan^(2)alpha+2^(2)tan^(2)alpha+...............+2^(n-1)tan2^(n-1)alpha+2^(n)cot 2^(n)alpha=cotalpha`