Problem 4

Prove the following equality for any natural number

n

and any real number

x

for which no denominator vanishes (that is,

x\ne \frac{k\pi}{2^t}

for

t=0,1,\dots,n

and integer

k

\frac{1}{\sin 2x}+\frac{1}{\sin 4x}+\frac{1}{\sin 8x}+\cdots+\frac{1}{\sin 2^nx} = \cot x - \cot 2^nx.

Step 5 of 5: State the identity for all

n

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In plain words

Because the argument in steps 1–3 never assumed a specific value of $n$ , the same computation proves the identity simultaneously for all $n$ , with no separate induction needed.

\frac{1}{\sin 2x}+\frac{1}{\sin 4x}+\cdots+\frac{1}{\sin 2^nx} = \cot x - \cot 2^nx

Detailed analysis

Steps 1–4 hold for every natural number $n$ and every admissible $x$ , so the telescoping identity of step 3 is exactly the equality that had to be proved.