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The value of 1 + c o s π 8 1 + c o s 3 π 8 1 + c o s 5 π 8 1 + c o s 7 π 8 is equal to

Options

  1. A1 2
  2. B1 4
  3. C1 8
  4. D1 16

Correct answer

C. 1 8

Step-by-step solution

Given expression can be written as 1 + cos ⁡ π 8 1 + cos ⁡ 7 π 8 1 + cos ⁡ 3 π 8 1 + cos ⁡ 5 π 8 = 1 + cos π 8 1 - cos π 8 1 + cos 3 π 8 1 - cos 3 π 8 = 1 - cos 2 π 8 1 - cos 2 3 π 8 = sin 2 π 8 sin 2 3 π 8 = sin 2 π 8 · cos 2 π 8 = 1 4 2 sin π 8 cos π 8 2 = 1 4 sin 2 π 4 = 1 8

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