Prove the following by the principle of mathematical induction: `1/2tan(x/2)+1/4tan(x/4)++1//2^ntan(x/(2^n))=1/(2^n)cot(x/(2^n))-cot x` for all `n in N` and 0

Explanation

We need to prove `frac{1}{2} tan (frac{x}{2})+frac{1}{4} tan (frac{x}{4})+ldots+frac{1}{2^{n}} tan (frac{x}{2^{n}})=frac{1}{2^{n}} cot (frac{x}{2^{n}})-cot x` for all `n in N` and `0
using mathematical induction.

For `n=1`,

LHS `=frac{1}{2} tan frac{x}{2}` and

`RHS =frac{1}{2} cot frac{x}{2}-cot x=frac{1}{2 tan frac{x}{2}}-frac{1}{tan x}`

`Rightarrow R H S=frac{1}{2 tan frac{x}{2}}-frac{1}{frac{2 tan frac{x}{2}}{1-tan ^{2} frac{4}{2}}} `

`Rightarrow R H S=frac{1}{2 tan frac{x}{2}}-frac{1-tan ^{2} frac{x}{2}}{2 tan frac{x}{2}}=frac{1-1+tan ^{2} frac{x}{2}}{2 tan frac{x}{2}}=frac{tan frac{x}{2}}{2}`

Therefore, the given relation is true for `n=1`.

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