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Question Number 185320 by mathlove last updated on 20/Jan/23

Answered by SEKRET last updated on 20/Jan/23

$$\mathbf{sin}\left(2\mathbf{a}\right)=2\mathbf{sin}\left(\mathbf{a}\right)\mathbf{cos}\left(\mathbf{a}\right)$$

Answered by SEKRET last updated on 20/Jan/23

$$\mathbf{sin}\left(\frac{\mathit{\pi }}{2^{10}}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^{10}}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot ........\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=\mathbf{A}$$$$2\cdot \mathbf{sin}\left(\frac{\mathit{\pi }}{2^{10}}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^{10}}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot ....\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=2\mathit{A}$$$$\mathbf{sin}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^8}\right)\cdot ....\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=2\mathbf{A}$$$$2\mathbf{sin}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^9}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^8}\right)....\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=2^2\cdot \mathbf{A}$$$$2\cdot \mathbf{sin}\left(\frac{\mathit{\pi }}{2^8}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^8}\right).....\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=2^3\mathbf{A}$$$$...........$$$$.................$$$$........................$$$$2\mathbf{sin}\left(\frac{\mathit{\pi }}{2^2}\right)\cdot \mathbf{cos}\left(\frac{\mathit{\pi }}{2^2}\right)=2^9\cdot \mathbf{A}$$$$\mathbf{sin}\left(\frac{\mathit{\pi }}{2}\right)=2^9\cdot \mathbf{A}\left[\begin{array}{c}\mathbf{A}=\frac{1}{2^9}\\ \mathit{A}=\frac{1}{512}\end{array}\right]$$$$\mathit{A}\mathit{B}\mathit{D}\mathit{U}\mathit{L}\mathit{A}\mathit{Z}\mathit{I}\mathit{Z}\mathit{A}\mathit{B}\mathit{D}\mathit{U}\mathit{V}\mathit{A}\mathit{L}\mathit{I}\mathit{Y}\mathit{E}\mathit{V}$$

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