Question

Hello! Please help!! I really need this.​

Answer 1

Lets show that:

• cos x cos 2x cos 4x cos 8x = sin 16x / 16 sinx

Use formula:

• sin 2x = 2 sin x cos x

Multiply LHS by 2sinx/2sinx:

• 2sinx cos x cos 2x cos 4x cos 8x / 2 sin x =
• sin 2x cos 2x cos 4x cos 8x / 2 sin x =
• 2sin 2x cos 2x cos 4x cos 8x / 4 sin x =
• sin 4x cos 4x cos 8x / 4 sin x =
• 2sin 4x cos 4x cos 8x / 8 sin x =
• sin 8x cos 8x / 8 sin x =
• 2 sin 8x cos 8x / 16 sin x =
• sin 16x / 16 sin x

Now, we can easily find that:

• sin (16*2π/15) = sin (2π/15)

Coming back to the original equation, we get:

• cos (2π/15) cos (4π/15) cos (8π/15) cos (16π/15) = sin (16*2π/15) / 16 sin (2π/15)
• cos (2π/15) cos (4π/15) cos (8π/15) cos (16π/15) = 1/16