Question

If Sin x not equal to 0,prove that cos x*Cos2x*Cos4x*cos8x=sin(2^4x)/2^4sinx and hence prove that cos 2π/15*cos4π/15*cos8π/15*cos14π/15=1/16​

Answer 1

Answer:

cos 2π/15 cos 4π/15 cos 8π/15 cos 16π/15

= cos(2π/15) cos(4π/15) cos(8π/15) cos(π + π/15)

= cos(2π/15) cos(4π/15) cos(8π/15) [-cos(π/15)]

= -[cos(4π/15) cos(π/15)] [cos(8π/15) cos(2π/15)]

= -(1/2)[2 cos 48° cos 12°] (1/2)[2 cos 96° cos 24°]

Using the formula 2 cos A cos B = cos(A + B) + cos(A – B),

= -(1/4)[cos(48° + 12°) + cos(48° – 12°)] [cos(96° + 24°) + cos(96° – 24°)]

= -(1/4)(cos 60° + cos 36°)(cos 120° + cos 72°)

= -(1/4) [(1/2) + (√5 + 1)/4] [(-1/2) + (√5 – 1)/4]

= -(1/4) [(3 + √5)/4] [(-3 + √5)/4]

= -(5 – 9)/64

= -(-4)/64

= 1/16