The value of 2^22^32^102^10:

Question

The value of $\cos\dfrac\pi{2^2}\cdot\cos\dfrac\pi{2^3}\cdots\cos\dfrac\pi{2^{10}}\cdot\sin\dfrac\pi{2^{10}}$:

Accepted Answer

Multiplying and dividing by $\sin\dfrac\pi{2^2}$, and repeatedly using $\sin\theta\cos\theta=\tfrac12\sin2\theta$:

$$= \frac1{2}\cos\tfrac\pi{2^2}\cos\tfrac\pi{2^3}\cdots\cos\tfrac\pi{2^9}\sin\tfrac\pi{2^9} = \cdots = \frac1{2^8}\cos\tfrac\pi{2^2}\sin\tfrac\pi{2^2} = \frac1{2^9}\sin\tfrac\pi2 = \frac1{512}$$

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