How to solve this one I know this is related to differentiation but how to proceed with this??? Please give all steps so that it is easily understood.
Given that $\cos(x/2)\cos(x/4)\cos(x/8)\ldots=(\sin x)/x,$ prove that $(1/2^2)\sec^2(x/2)+(1/2^4)\sec^2(x/4)\ldots=\csc^2(x) - (1/x^2).$
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derivatives
1 Answers
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Taking logs of both sides, $$ \sum_{k=1}^{\infty} \log{\cos{\left( \frac{x}{2^k} \right)}} = \log{\sin{x}}-\log{x}. $$ Differentiating twice, $$ \begin{align} \sum_{k=1}^{\infty} -\frac{1}{2^k} \tan{\left(\frac{x}{2^k} \right)} &= \cot{x}-\frac{1}{x} \\ \sum_{k=1}^{\infty} -\frac{1}{2^{2k}} \sec^2{\left(\frac{x}{2^k} \right)} &= -\csc^2{x}+\frac{1}{x^2}. \end{align} $$
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1Good idea! But what if you have some $x$ such that $\cos (x/2^k)<0$ for some $k$? – 2017-01-03
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2The simplest thing to do for real $x$ would be to square both sides, then take logs and divide by $2$. The derivatives will be the same. For complex arguments, proving it on a neighbourhood of zero and applying meromorphic continuation is enough. – 2017-01-03