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Question Number 26999 by abdo imad last updated on 01/Jan/18

$$calculate\prod _{k=1}^{n}cos\left(\frac{a}{2^k}\right)and0<a<\pi thenfindthevalueof$$ $${lim}_{n->\propto }\sum _{k=1}^{n}ln\left(cos\left(\frac{a}{2^k}\right)\right).$$

Answered by prakash jain last updated on 01/Jan/18

$$\underset{k=1}{\prod ^n}\cos \left(\frac{a}{2^k}\right)$$ $$=\frac{1}{\sin \left(\frac{a}{2^n}\right)}\sin \left(\frac{a}{2^n}\right)\underset{k=1}{\prod ^n}\cos \left(\frac{a}{2^k}\right)$$ $$=\frac{\sin a}{2^n\sin \left(\frac{a}{2^n}\right)}$$ $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sin a}{2^n\sin \left(\frac{a}{2^n}\right)}$$ $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sin a}{\frac{\sin \left(\frac{a}{2^n}\right)}{\frac{a}{2^7}}\cdot a}=\frac{\sin a}{a}$$ $$$$

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