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Value of a product of sines or cosines of special angles

This type of question, asked 7 times: 2021 Paper 19, 26 Jun 2022, 25 Jul 2022, 8 Apr 2023, 10 Apr 2023, 24 Jan 2026 S1, 2 Apr 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 19

    73
    The value of 2sin⁡(π8)sin⁡(2π8)sin⁡(3π8)sin⁡(5π8)sin⁡(6π8)sin⁡(7π8)2\sin\left( \frac{\pi}{8} \right)\sin\left( \frac{2\pi}{8} \right)\sin\left( \frac{3\pi}{8} \right)\sin\left( \frac{5\pi}{8} \right)\sin\left( \frac{6\pi}{8} \right)\sin\left( \frac{7\pi}{8} \right) is :

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  2. Question 2

    26 Jun 2022

    86
    If sin⁡2(10∘)sin⁡(20∘)sin⁡(40∘)sin⁡(50∘)sin⁡(70∘)=α−\sin^{2}\left( 10^{\circ} \right)\sin\left( 20^{\circ} \right)\sin\left( 40^{\circ} \right)\sin\left( 50^{\circ} \right)\sin\left( 70^{\circ} \right)=\alpha- 116sin⁡(10∘)\frac{1}{16}\sin\left( 10^{\circ} \right), then 16+α−116 +\alpha^{- 1} is equal to

  3. Question 3

    26 Jun 2022

    168
    16sin⁡(20∘)sin⁡(40∘)sin⁡(80∘)16\sin\left( 20^{\circ} \right)\sin\left( 40^{\circ} \right)\sin\left( 80^{\circ} \right) is equal to:

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  4. Question 4

    25 Jul 2022

    169
    2sin⁡(π22)sin⁡(3π22)sin⁡(5π22)sin⁡(7π22)sin⁡(9π22)2\sin\left( \frac{\pi}{22} \right)\sin\left( \frac{3\pi}{22} \right)\sin\left( \frac{5\pi}{22} \right)\sin\left( \frac{7\pi}{22} \right)\sin\left( \frac{9\pi}{22} \right) is equal to

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  5. Question 5

    8 Apr 2023

    159
    The value of 36(4cos⁡29∘−1)(4cos⁡227∘−1)(4cos⁡281∘−1)(4cos⁡2243∘−1)36\left( 4\cos^{2}9^{\circ}- 1 \right)\left( 4\cos^{2}27^{\circ}- 1 \right)\left( 4\cos^{2}81^{\circ}- 1 \right)\left( 4\cos^{2}243^{\circ}- 1 \right) is

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  6. Question 6

    10 Apr 2023

    68
     96cos⁡π33cos⁡2π33cos⁡4π33cos⁡8π33cos⁡16π33\ 96\cos\frac{\pi}{33}\cos\frac{2\pi}{33}\cos\frac{4\pi}{33}\cos\frac{8\pi}{33}\cos\frac{16\pi}{33} equal to

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  7. Question 7

    24 Jan 2026 S1

    58
    The value of 3csc⁡20∘−sec⁡20∘cos⁡20∘cos⁡40∘cos⁡60∘cos⁡80∘\frac{\sqrt{3}\csc20^{\circ}- \sec20^{\circ}}{\cos20^{\circ}\cos40^{\circ}\cos60^{\circ}\cos80^{\circ}} is equal to

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  8. Question 8

    2 Apr 2026 S1

    61
    If sin⁡(π18)sin⁡(5π18)sin⁡(7π18)=K\sin\left( \frac{\pi}{18} \right)\sin\left( \frac{5\pi}{18} \right)\sin\left( \frac{7\pi}{18} \right)= K, then the value of sin⁡(10Kπ3)\sin\left( \frac{10K\pi}{3} \right) is :

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