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Composition of Trigonometric Functions and Inverse Trigonometric Functions

Find the valu...

Question

Find the value of cos (pi/13) × cos(2pi/13) × cos (3pi/13) × cos (4pi/13) × cos (5pi/13) × cos ( 6pi/13) (A) 1/64 (B) -1/64 (C) 1/32 (D) -1/8

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Solution

$Given equation is, \cos (\frac{π}{13}) \cos (\frac{2 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{4 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13}) Now multiply and divide by 2 \sin (\frac{π}{13}) = \frac{2 \sin (\frac{π}{13}) \cos (\frac{π}{13}) \cos (\frac{2 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{4 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13})}{2 \sin (\frac{π}{13})} Since \sin 2 a = 2 sinacosa we get, = \frac{\sin (\frac{2 π}{13}) \cos (\frac{2 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{4 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13})}{2 \sin (\frac{π}{13})} = \frac{2 \sin (\frac{2 π}{13}) \cos (\frac{2 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{4 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13})}{2^{2} \sin (\frac{π}{13})} = \frac{2 \sin (\frac{4 π}{13}) \cos (\frac{4 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13})}{2^{3} \sin (\frac{π}{13})} = \frac{\sin (\frac{8 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13})}{2^{3} \sin (\frac{π}{13})} Now divide and multiply by 2 \sin (\frac{3 π}{13}) = \frac{2 \sin (\frac{3 π}{13}) \cos (\frac{3 π}{13}) \cos (\frac{5 π}{13}) \cos (\frac{6 π}{13}) \sin (\frac{8 π}{13})}{2^{4} \sin (\frac{π}{13}) \sin (\frac{3 π}{13})} = \frac{2 \sin (\frac{6 π}{13}) \cos (\frac{6 π}{13}) \sin (\frac{8 π}{13}) \cos (\frac{5 π}{13})}{2^{5} \sin (\frac{π}{13}) \sin (\frac{3 π}{13})} = \frac{\sin (\frac{12 π}{13}) \sin (\frac{8 π}{13}) \cos (\frac{5 π}{13})}{2^{5} \sin (\frac{π}{13}) \sin (\frac{3 π}{13})} = \frac{\sin (π - \frac{π}{13}) \sin (π - \frac{5 π}{13}) \cos (\frac{5 π}{13})}{2^{5} \sin (\frac{π}{13}) \sin (\frac{3 π}{13})} = \frac{\sin (\frac{π}{13}) \sin (\frac{5 π}{13}) \cos (\frac{5 π}{13})}{2^{5} \sin (\frac{π}{13}) \sin (\frac{3 π}{13})} = \frac{2 \sin (\frac{5 π}{13}) \cos (\frac{5 π}{13})}{2^{6} \sin (π - \frac{10 π}{13})} = \frac{\sin (\frac{10 π}{13})}{2^{6} \sin (\frac{10 π}{13})} = \frac{1}{64} =$

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