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Cos π/7*Cos 2π/7*Cos 10π/7 – Sin π/14*Sin 3π/14*Sin 5π/14 =
E =sin(π/14) sin(3π/14) sin(5π/14) Put : x = π/14. Then : 14x = π. ∴ E = sin x. sin 3x, sin 5x. ∴ E = sin (π/14). sin (3π/14). sin (5π/14) ......... (1) Now, sin π/14 = sin (π/2 - 6π/14) = cos 6π/14 = cos (π - 8π/14) = - cos 8π/14 = - cos 8x Similarly, sin 3π/14 = cos 4π/14 = cos 4x and sin 5π/14 = cos 2π/14 = cos 2x. ∴ from (1), E = - cos 8x. cos 4x. cos 2x Multiply and divide by 2 sin2x, ...= -1/ (2 sin2x ). [( 2 sin 2x. cos 2x ). cos 4x. cos 8x] ...= (-1/ 4 sin 2x ). ( 2 sin 4x. cos 4x ). cos 8x ...= (-1/ 8 sin 2x ). ( 2 sin 8x. cos 8x ) ...= (-1/ 8 sin 2x ). sin 16x ...= (-1/8). ( 1/ (sin π/7)). sin 8π/7 ...= (-1/8). ( 1/ (sin π/7)). sin ( π + π/7 ) ...= - (1/8). ( 1/ sin π/7 ). ( - sin π/7 ) ...= 1/8. ∴ E = 1/8 is the answer
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