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Exact value of an expression in sin 18, cos 36 and related angles

This type of question, asked 5 times: 2021 Paper 22, 25 Jun 2022, 27 Jun 2022, 8 Apr 2024, 22 Jan 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 22

    72
    csc⁡18∘\csc18^{\circ} is a root of the equation :

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  2. Question 2

    25 Jun 2022

    165
    The value of 2sin⁡(12∘)−sin⁡(72∘)2\sin\left( 12^{\circ} \right)- \sin\left( 72^{\circ} \right) is :

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  3. Question 3

    27 Jun 2022

    169
    α=sin⁡36∘\alpha = \sin36^{\circ} is a root of which of the following equations?

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  4. Question 4

    8 Apr 2024

    159
    If the value of 3cos⁡36∘+5sin⁡18∘5cos⁡36∘−3sin⁡18∘\frac{3\cos36^{\circ}+ 5\sin18^{\circ}}{5\cos36^{\circ}- 3\sin18^{\circ}} is a5−bc\frac{a\sqrt{5}-b}{c}, where a,b,ca,b,c are natural numbers and gcd⁡(a,c)=1\gcd(a,c) = 1, then a+b+ca + b + c is equal to:

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  5. Question 5

    22 Jan 2026 S1

    74
    If cos⁡248∘−sin⁡212∘sin⁡224∘−sin⁡26∘=α+β52\frac{\cos^{2}48^{\circ} - \sin^{2}12^{\circ}}{\sin^{2}24^{\circ} - \sin^{2}6^{\circ}} = \frac{\alpha + \beta\sqrt{5}}{2}, where α,β∈N\alpha,\beta \in \mathbb{N}, then α+β\alpha + \beta is equal to ____\_\_\_\_ .