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  • True and False cos⁡〖2π/15〗 cos⁡〖4π/15〗 cos⁡〖8π/15〗 cos⁡〖16π/15〗 = 1/16

True and False

\cos \frac{2 \pi }{15}\cos \frac{4 \pi }{15}\cos \frac{8 \pi }{15}\cos \frac{16 \pi }{15} = \frac{1}{16}

Answers (1)

\\\cos \frac{2 \pi }{15}\cos \frac{4 \pi }{15}\cos \frac{8 \pi }{15}\cos \frac{16 \pi }{15} \\\\ =\cos 24\cos 48\cos 96\cos 192 \\\\ =\frac{1}{16\sin 24} \left( 2\sin 24\cos 24 \right) \left( 2\cos 48 \right) \left( 2\cos 96 \right) \left( 2\cos 192 \right) \\\\ =\frac{1}{16\sin 24} \left( 2\sin 48\cos 48 \right) \left( 2\cos 96 \right) \left( 2\cos 192 \right) \\\\ =\frac{1}{16\sin 24} \left( 2\cos 96\sin 96 \right) \left( 2\cos 192 \right) =\frac{2\sin 192\cos 192}{16\sin 24} \\\\ =\frac{\sin 384}{16\sin 24}=\frac{\sin \left( 360+24 \right) }{16\sin 24} \\\\ =\frac{\sin 24}{16\sin 24}=\frac{1}{16} \\\\

Hence, the statement is ‘true’. 

 

Posted by

infoexpert21

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