Given that $\cos \frac{x}{2}.\cos \frac{x}{4}.\cos \frac{x}{8}.......=\Large\frac {\sin x}{x},$ $$Prove\;that\;\frac{1}{2^2}\sec^2\frac{x}{2}+\frac{1}{2^4}\sec^2\frac{x}{4}+.......=cosec^2x-\frac{1}{x^2}$$

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